🧠:<think>Alright, so I have this problem about Alex and Betty meeting for coffee. Let me try to figure this out step by step. First, the problem is divided into two parts. The first part is about calculating the expected number of free coffees Alex would receive in a year, given his Poisson-distributed visits. The second part is about finding the joint probability distribution of their visits in a week and then calculating the expected number of times they meet in a month.Starting with the first part. Alex visits the café following a Poisson process with an average rate of 5 visits per week. The café has a loyalty program where every 10th coffee is free. So, I need to find out how many free coffees Alex can expect in a year, which is 52 weeks.Hmm, okay. So, since Alex's visits are Poisson-distributed, the number of coffees he buys in a week is a Poisson random variable with λ = 5. But wait, the loyalty program is based on the total number of coffees he buys, right? So, every 10th coffee is free. That means for every 10 coffees he buys, he gets one free. So, the number of free coffees he gets is essentially the number of times he reaches a multiple of 10 in his total visits.But wait, is it based on the total number of coffees he buys over the year, or is it based on each week? The problem says "given his Poisson-distributed visits," and "Alex never misses a free coffee due to his irregular visits." So, I think it's based on the total number of coffees he buys in a year. So, the loyalty program is cumulative, meaning that every 10th coffee he buys, regardless of when, gives him a free one.So, to find the expected number of free coffees, I need to find the expected number of coffees Alex buys in a year and then divide that by 10, since every 10th coffee is free.First, let's calculate the expected number of coffees Alex buys in a year. Since his visits are Poisson with λ = 5 per week, the expected number per week is 5. Therefore, over 52 weeks, the expected number is 5 * 52 = 260 coffees.So, if he buys 260 coffees in a year, the number of free coffees he gets is 260 / 10 = 26. So, he can expect 26 free coffees in a year.Wait, but hold on. Is it that straightforward? Because the Poisson process counts the number of events in a fixed interval, which in this case is weekly. But the loyalty program is based on the total number of coffees, regardless of the week. So, actually, the total number of coffees he buys in a year is the sum of 52 independent Poisson random variables each with λ = 5. The sum of independent Poisson variables is also Poisson with λ equal to the sum of the individual λs. So, the total λ for a year is 5 * 52 = 260. So, the number of coffees he buys in a year is Poisson(260). Therefore, the expected number is 260, so the expected number of free coffees is 260 / 10 = 26.But wait, is the number of free coffees the floor division of total coffees divided by 10? Or is it the integer division? For example, if he buys 26 coffees, he gets 2 free. If he buys 27, he still gets 2 free. So, actually, the number of free coffees is the floor of (total coffees / 10). But since we're dealing with expectations, we can approximate it as total coffees divided by 10 because the expectation of the floor function is approximately the floor of the expectation when dealing with large numbers, which 260 is.Alternatively, maybe it's better to model it as the expectation of floor(N/10), where N is Poisson(260). But for large λ, the expectation of floor(N/10) is approximately λ / 10. So, 260 / 10 = 26. So, the expected number of free coffees is 26.Okay, that seems reasonable. So, part 1 answer is 26.Moving on to part 2. It says that Alex and Betty's visits overlap 60% of the time. I need to determine the joint probability distribution of their visits in a week and then calculate the expected number of times they meet in a month (4 weeks).First, let me parse this. Betty has a consistent schedule, exactly 4 times a week. So, Betty's visits are deterministic? Or is it that she has a consistent schedule, meaning maybe she has fixed days she visits, but the problem says "visits overlap 60% of the time." Hmm, maybe their visits overlap 60% of the time when they are both at the café.Wait, the problem says, "Given that Alex and Betty's visits overlap 60% of the time, determine the joint probability distribution of their visits in a week and then calculate the expected number of times they meet in a month (4 weeks)."So, perhaps we need to model their visits as some joint distribution where the probability that they are both at the café at the same time is 60%.Wait, but Alex's visits are Poisson with λ = 5 per week, and Betty's visits are exactly 4 per week. So, Betty's visits are fixed at 4 times a week, but Alex's are random.So, perhaps the overlapping probability is 60%, meaning that when Betty is at the café, there's a 60% chance Alex is also there. Or maybe it's the probability that their visits overlap is 60%.Wait, the problem says, "Given that Alex and Betty's visits overlap 60% of the time," so I think it means that when both are at the café, the probability that their visits overlap is 60%. Hmm, not entirely clear.Alternatively, maybe it's the probability that on any given day, if Betty is at the café, Alex is also there 60% of the time. Or maybe it's the overall probability that their visits coincide is 60%.Wait, perhaps it's better to model their visits as independent processes, but with some dependency.Wait, let's think about it. Betty visits exactly 4 times a week. So, if we model the week as having 7 days, Betty chooses 4 days to visit. Alex, on the other hand, has a Poisson process with λ = 5 per week. So, his visits are random, with an average of 5 per week.But the visits overlap 60% of the time. So, perhaps the probability that Alex and Betty are at the café on the same day is 60%.Wait, but Betty is at the café on 4 specific days each week. So, if we model Betty's visits as fixed, say on days 1, 2, 3, 4, then Alex's visits are Poisson distributed over the week.But the problem says their visits overlap 60% of the time. So, perhaps the probability that Alex is at the café on any given day is such that, considering Betty's fixed days, the probability that Alex is also there on those days is 60%.Wait, maybe we need to model the joint distribution where for each of Betty's visit days, the probability that Alex is also visiting is 60%. So, for each of Betty's 4 visits, there's a 60% chance Alex is also there.Alternatively, maybe the overall probability that their visits overlap is 60%, meaning that the expected number of overlapping visits per week is 60% of the minimum of their visits.Wait, I'm getting confused. Let me try to structure this.First, Betty visits exactly 4 times a week. Let's denote Betty's visits as fixed on 4 specific days. Alex's visits are Poisson with λ = 5 per week, so the number of visits Alex makes in a week is Poisson(5). The times of Alex's visits are uniformly distributed over the week.So, if we consider each day of the week, the probability that Alex visits on a specific day is 1 - e^{-λ/7}, where λ/7 is the average number of visits per day. Wait, actually, for a Poisson process, the number of events in disjoint intervals are independent, and the number of events in each interval follows a Poisson distribution with parameter equal to the rate multiplied by the interval length.So, if the weekly rate is 5, then the daily rate is 5/7. So, the probability that Alex visits on a specific day is 1 - e^{-5/7}.But Betty visits on 4 specific days. So, the probability that Alex visits on each of those days is 1 - e^{-5/7}. Therefore, the expected number of overlapping visits per week is 4 * (1 - e^{-5/7}).But the problem says that their visits overlap 60% of the time. So, perhaps 60% is the probability that they overlap on any given day when Betty is visiting.Wait, maybe it's the overall probability that their visits overlap is 60%, meaning that the expected number of overlapping visits per week is 60% of the minimum of their visits.Wait, I'm not sure. Let me try to think differently.If Betty visits exactly 4 times a week, and Alex's visits are Poisson(5) per week, then the expected number of overlapping visits per week is the sum over each of Betty's visit days of the probability that Alex visits on that day.So, if Betty visits on 4 specific days, and the probability that Alex visits on each day is p, then the expected number of overlapping visits is 4p.But we are told that their visits overlap 60% of the time. So, perhaps p = 0.6? Or maybe the expected number of overlapping visits is 60% of the maximum possible, which is 4.Wait, if the maximum possible overlapping visits is 4 (since Betty only visits 4 times), then 60% of 4 is 2.4. So, the expected number of overlapping visits per week is 2.4.But how is this related to the joint probability distribution?Alternatively, maybe the probability that they meet on any given day is 60%, so the joint probability distribution would have a 60% chance on the days Betty visits, and 0 otherwise.Wait, perhaps we need to model their visits as a Bernoulli process where on each day Betty visits, there's a 60% chance Alex also visits.So, if Betty visits on 4 days, then on each of those days, the probability that Alex is also there is 0.6. So, the number of overlapping visits per week is a binomial random variable with n=4 and p=0.6.Therefore, the expected number of overlapping visits per week is 4 * 0.6 = 2.4.Then, over a month (4 weeks), the expected number of overlapping visits would be 2.4 * 4 = 9.6.But wait, let me make sure.Alternatively, perhaps the joint probability distribution is such that for each of Betty's visits, the probability that Alex is also visiting is 0.6. So, the number of meetings per week is a binomial(4, 0.6) random variable, with expectation 2.4.Therefore, over 4 weeks, the expectation is 2.4 * 4 = 9.6.But let me think again. The problem says, "determine the joint probability distribution of their visits in a week." So, maybe we need to model the joint distribution where Betty has 4 visits, and for each of those, Alex has a 60% chance of being there.So, the joint distribution would be that Betty's visits are fixed at 4, and for each of those, Alex's presence is a Bernoulli trial with p=0.6.Therefore, the number of meetings per week is a binomial(4, 0.6) variable, with expectation 2.4.Thus, over 4 weeks, the expected number is 2.4 * 4 = 9.6.But the problem says "calculate the expected number of times they meet in a month (4 weeks)." So, 9.6 is the expectation.But let me check if this is the correct approach.Alternatively, since Alex's visits are Poisson(5) per week, and Betty's are fixed at 4, the expected number of overlapping visits is the sum over each of Betty's visit days of the probability that Alex visits on that day.So, if Betty visits on 4 days, and the probability that Alex visits on each day is p, then the expected number of overlapping visits is 4p.But we are told that their visits overlap 60% of the time. So, perhaps p = 0.6.Therefore, the expected number of overlapping visits per week is 4 * 0.6 = 2.4, as before.Hence, over 4 weeks, it's 2.4 * 4 = 9.6.So, the expected number of times they meet in a month is 9.6, which is 9.6 ≈ 9.6.But since we're dealing with expectations, we can keep it as 9.6.Alternatively, if we model the joint distribution as independent, but with a 60% overlap, perhaps it's a different approach.Wait, another way to think about it is that the probability that Alex and Betty are at the café at the same time is 60%. So, the joint probability distribution would have a 60% chance that they meet on any given day when Betty is visiting.But since Betty visits 4 times a week, the expected number of meetings is 4 * 0.6 = 2.4 per week, leading to 9.6 per month.Yes, that seems consistent.So, putting it all together:1. Expected free coffees for Alex in a year: 26.2. Expected number of meetings in a month: 9.6.But let me double-check the first part again.Alex's visits are Poisson(5) per week, so over 52 weeks, it's Poisson(260). The number of free coffees is floor(260 / 10) = 26. But since we're dealing with expectation, it's exactly 260 / 10 = 26.Yes, that's correct.For the second part, since Betty visits 4 times a week, and each visit has a 60% chance of overlapping with Alex, the expected number of meetings per week is 4 * 0.6 = 2.4. Over 4 weeks, it's 2.4 * 4 = 9.6.So, the answers are 26 and 9.6.But wait, the problem says "determine the joint probability distribution of their visits in a week." So, perhaps I need to model it more formally.Let me denote Betty's visits as fixed at 4 days per week. Let’s say she visits on days D1, D2, D3, D4. Alex's visits are Poisson(5) per week, so the number of visits on each day is Poisson(5/7), approximately.But the problem states that their visits overlap 60% of the time. So, perhaps the probability that Alex visits on any of Betty's visit days is 0.6.Wait, but if Betty visits on 4 days, and the probability that Alex visits on each of those days is p, then the expected number of overlapping visits is 4p.Given that the expected number of overlapping visits is 60% of the maximum possible, which is 4, so 4 * 0.6 = 2.4. So, p = 0.6.Therefore, the joint probability distribution is such that on each of Betty's 4 visit days, Alex has a 60% chance of also visiting. So, the number of overlapping visits per week is a binomial(4, 0.6) random variable.Hence, the expected number of meetings per week is 2.4, leading to 9.6 per month.Yes, that seems correct.So, summarizing:1. Alex's expected free coffees in a year: 26.2. Expected number of meetings in a month: 9.6.But wait, 9.6 is a decimal. Should we round it? The problem doesn't specify, so probably keep it as 9.6.Alternatively, if we think in terms of exact expectation, it's 24/2.5 = 9.6.Yes, so 9.6 is correct.So, final answers:1. 262. 9.6But let me write them in boxed notation as requested.Final Answer1. The expected number of free coffees Alex would receive in a year is boxed{26}.2. The expected number of times Alex and Betty meet in a month is boxed{9.6}.</think>

🧠:<think>Okay, so I have this problem that my colleague came up with, inspired by memes. It's got two parts, and I need to figure out both. Let me take it step by step.Starting with Sub-problem 1: The colleague wants to create 5 distinct memes, each made from one image and one caption. There are 12 unique images and 8 unique captions. I need to find out how many ways they can create this set of 5 distinct memes.Hmm, so each meme is a combination of one image and one caption. Since the images and captions are unique, each meme is a unique pair. So, for each meme, there are 12 choices for the image and 8 choices for the caption. That means for one meme, there are 12 * 8 = 96 possible combinations.But wait, the colleague wants to create 5 distinct memes. So, I think this is a problem of selecting 5 unique pairs from the total possible pairs. Since each meme is a unique combination, the total number of possible memes is 12 * 8 = 96, as I just calculated.So, the number of ways to choose 5 distinct memes from 96 is a combination problem, right? Because the order in which the memes are created doesn't matter; it's just the set of 5 that matters.So, the formula for combinations is C(n, k) = n! / (k! * (n - k)!), where n is the total number, and k is the number we want to choose.Plugging in the numbers, n is 96 and k is 5. So, the number of ways is C(96, 5). Let me compute that.But wait, hold on. Is it that straightforward? Because each meme is a unique image-caption pair, but when creating 5 memes, do we have to consider that each image can only be used once and each caption can only be used once? Or can they be reused?Looking back at the problem statement: "create a set of 5 distinct memes, each meme consisting of one image and one caption." It says "distinct" memes, but it doesn't specify whether the images or captions have to be unique across the set. Hmm.Wait, actually, the images and captions themselves are unique. So, each image can be used in multiple memes, but since the memes have to be distinct, each meme must be a unique combination. So, for example, if I use image 1 with caption 1, I can't use image 1 with caption 1 again, but I can use image 1 with caption 2, as long as that's a different meme.But wait, actually, the problem says "5 distinct memes." So, each meme is unique, so each image-caption pair must be unique. So, in that case, the total number of possible memes is 12 * 8 = 96, and we need to choose 5 distinct ones. So, yes, it's just C(96, 5).But hold on, another thought: sometimes in these problems, when you have pairs, you have to consider whether you can reuse images or captions. For example, if each image can only be used once, then it's a different problem. Similarly for captions.But the problem doesn't specify that images or captions can't be reused. It just says "5 distinct memes," each consisting of one image and one caption. So, I think that means that the meme pairs have to be unique, but the images and captions themselves can be repeated across different memes. So, for example, image 1 can be used in multiple memes as long as each time it's paired with a different caption.Wait, but if that's the case, then the total number of possible memes is 12 * 8 = 96, and we're choosing 5 distinct ones, so the number of ways is C(96, 5).But let me think again. If images and captions can be reused, then the number of possible memes is indeed 96, and choosing 5 is C(96,5). But if the images and captions can't be reused, meaning each image can be used only once and each caption only once, then it's a different calculation.Wait, the problem says "create a set of 5 distinct memes, each meme consisting of one image and one caption." So, it's about the memes being distinct, not necessarily the images or captions. So, I think the first interpretation is correct: the total number of possible memes is 96, and we need to choose 5 distinct ones. So, the number of ways is C(96, 5).But just to be thorough, let's consider the other scenario where each image can be used only once and each caption only once. Then, it's a problem of selecting 5 images out of 12 and 5 captions out of 8, and then pairing them. So, the number of ways would be C(12,5) * C(8,5) * 5!.Wait, because you choose 5 images, 5 captions, and then assign each image to a caption, which is 5! permutations.So, in that case, the number would be C(12,5) * C(8,5) * 5!.But which interpretation is correct? The problem says "create a set of 5 distinct memes, each meme consisting of one image and one caption." It doesn't specify whether the images or captions have to be unique across the set. So, it's ambiguous.Wait, but in the context of creating memes, it's common to have multiple memes with the same image but different captions, or same caption with different images. So, I think the first interpretation is more likely intended: that the memes are distinct, but images and captions can be reused. So, the number is C(96,5).But just to be safe, maybe the problem expects the second interpretation, where each image and caption can be used only once. So, let's compute both.First interpretation: C(96,5). Let's compute that.96 choose 5 is 96! / (5! * 91!) = (96 * 95 * 94 * 93 * 92) / (5 * 4 * 3 * 2 * 1).Calculating that:96 * 95 = 91209120 * 94 = 856, 80? Wait, 9120 * 94:Let me compute 9120 * 90 = 820,8009120 * 4 = 36,480So total is 820,800 + 36,480 = 857,280Then, 857,280 * 93:Hmm, that's a big number. Maybe I should compute step by step.Wait, actually, maybe it's better to compute 96 * 95 * 94 * 93 * 92 first, then divide by 120.But let me see:96 * 95 = 91209120 * 94 = 857,280857,280 * 93: Let's compute 857,280 * 90 = 77,155,200857,280 * 3 = 2,571,840Total: 77,155,200 + 2,571,840 = 79,727,04079,727,040 * 92: Hmm, this is getting too big. Maybe I should just note that 96 choose 5 is 7,528,752.Wait, actually, I think 96 choose 5 is 7,528,752. Let me verify:Yes, 96C5 is calculated as 96! / (5! * 91!) = (96 × 95 × 94 × 93 × 92) / (5 × 4 × 3 × 2 × 1) = (96 × 95 × 94 × 93 × 92) / 120.Calculating numerator:96 × 95 = 91209120 × 94 = 857,280857,280 × 93 = 79,727,04079,727,040 × 92 = 7,334, let me compute 79,727,040 × 90 = 7,175,433,60079,727,040 × 2 = 159,454,080Total: 7,175,433,600 + 159,454,080 = 7,334,887,680Now divide by 120:7,334,887,680 / 120 = 61,124,064Wait, that can't be right because 96 choose 5 is known to be 7,528,752. So, I must have messed up the multiplication somewhere.Wait, maybe I should compute it step by step:First, 96 × 95 = 91209120 × 94 = 9120 × 90 + 9120 × 4 = 820,800 + 36,480 = 857,280857,280 × 93 = 857,280 × 90 + 857,280 × 3 = 77,155,200 + 2,571,840 = 79,727,04079,727,040 × 92 = 79,727,040 × 90 + 79,727,040 × 2 = 7,175,433,600 + 159,454,080 = 7,334,887,680Now, 7,334,887,680 divided by 120:Divide by 10: 733,488,768Divide by 12: 733,488,768 / 12 = 61,124,064Wait, but I thought 96C5 is 7,528,752. Maybe I'm wrong. Let me check online.Wait, no, I think I made a mistake in the calculation. Let me try another approach.Compute 96C5:96C5 = (96 × 95 × 94 × 93 × 92) / (5 × 4 × 3 × 2 × 1)Compute numerator step by step:96 × 95 = 91209120 × 94 = 857,280857,280 × 93 = 79,727,04079,727,040 × 92 = 7,334,887,680Denominator: 5 × 4 × 3 × 2 × 1 = 120So, 7,334,887,680 / 120 = 61,124,064Wait, but I think I'm wrong because 96C5 is actually 7,528,752. Maybe I'm confusing it with something else.Wait, let me compute 96C5 using another method. Let's compute it as:96C5 = 96! / (5! * 91!) = (96 × 95 × 94 × 93 × 92) / (5 × 4 × 3 × 2 × 1)Let me compute numerator:96 × 95 = 91209120 × 94 = 857,280857,280 × 93 = 79,727,04079,727,040 × 92 = 7,334,887,680Denominator: 120So, 7,334,887,680 / 120 = 61,124,064Wait, but I think I'm wrong because 96C5 is actually 7,528,752. Maybe I'm confusing it with something else. Let me check with a calculator.Wait, no, I think I'm correct. Maybe 96C5 is indeed 61,124,064. Let me verify with a smaller number. For example, 10C5 is 252, which is correct. 20C5 is 15504. So, 96C5 being 61 million seems plausible.Wait, but let me think again. 96C5 is the number of ways to choose 5 items from 96, which is indeed 61,124,064.But wait, in the problem, each meme is a unique image-caption pair. So, if images and captions can be reused, then yes, it's 96C5. But if they can't be reused, then it's a different number.Wait, let's consider the second interpretation where each image and caption can be used only once. So, we need to choose 5 unique images from 12 and 5 unique captions from 8, and then pair them.So, the number of ways is C(12,5) * C(8,5) * 5!.Because:- C(12,5): choose 5 images from 12.- C(8,5): choose 5 captions from 8.- 5!: assign each chosen image to a chosen caption.So, let's compute that.C(12,5) = 792C(8,5) = 565! = 120So, total number of ways is 792 * 56 * 120Compute 792 * 56 first:792 * 50 = 39,600792 * 6 = 4,752Total: 39,600 + 4,752 = 44,352Now, 44,352 * 120:44,352 * 100 = 4,435,20044,352 * 20 = 887,040Total: 4,435,200 + 887,040 = 5,322,240So, if images and captions can't be reused, the number is 5,322,240.But which one is correct? The problem says "create a set of 5 distinct memes, each meme consisting of one image and one caption." It doesn't specify whether the images or captions have to be unique across the set. So, I think the first interpretation is correct, that the memes are distinct, but images and captions can be reused. So, the number is 61,124,064.But I'm a bit confused because sometimes in combinatorics, when you have pairs, you assume that each element is used only once. But in this case, the problem doesn't specify that, so I think the first interpretation is correct.So, Sub-problem 1 answer is 61,124,064.Wait, but let me think again. If the colleague is creating 5 distinct memes, each consisting of one image and one caption, and the images and captions are unique, does that mean that each image can be used only once and each caption only once? Or can they be used multiple times?The problem says "unique images and captions," but when creating the memes, it's about the memes being distinct. So, if the images and captions are unique, but the memes are just pairs, then the same image can be paired with different captions, and the same caption can be paired with different images. So, the total number of possible memes is 12 * 8 = 96, and choosing 5 distinct ones is C(96,5).Yes, I think that's correct.Okay, moving on to Sub-problem 2: The colleague randomly distributes these 5 distinct memes among 10 colleagues, each receiving at most one meme. What's the probability that exactly 3 specific colleagues (A, B, and C) each receive one meme, while the remaining 2 memes are given to any of the remaining 7 colleagues.So, we have 10 colleagues, 5 memes. Each colleague can receive at most one meme. So, it's like distributing 5 distinct memes to 10 distinct colleagues, each getting at most one.We need the probability that exactly A, B, and C each get one meme, and the other two memes go to any of the remaining 7 colleagues.So, probability is (number of favorable distributions) / (total number of possible distributions).First, let's compute the total number of possible distributions.Since each meme is distinct and each colleague can receive at most one, it's equivalent to choosing 5 colleagues out of 10 and assigning each a unique meme.So, the total number of ways is P(10,5) = 10 * 9 * 8 * 7 * 6.Alternatively, it's 10 choose 5 multiplied by 5!, which is the same as 10P5.Compute 10P5: 10! / (10-5)! = 10! / 5! = 30240.So, total number of possible distributions is 30,240.Now, the number of favorable distributions: exactly A, B, and C each receive one meme, and the other two memes go to any of the remaining 7 colleagues.So, we need to count the number of ways where A, B, and C each get one meme, and the other two memes go to the remaining 7 colleagues (so, not A, B, or C).So, let's break it down:1. Assign one meme to A, one to B, one to C.2. Assign the remaining two memes to any of the remaining 7 colleagues.But we also need to consider the assignment of specific memes to specific people.So, the total number of favorable distributions is:- Choose 3 memes out of 5 to assign to A, B, and C.- Assign these 3 memes to A, B, and C.- Assign the remaining 2 memes to any of the remaining 7 colleagues.Wait, but actually, since the memes are distinct, the assignment matters.Alternatively, think of it as:- Assign 3 specific memes to A, B, and C, and the other 2 to the remaining 7.But actually, it's more precise to think:- Choose 3 memes out of 5 to give to A, B, and C.- Assign these 3 memes to A, B, and C in some order.- Assign the remaining 2 memes to any of the remaining 7 colleagues in some order.So, let's compute that.First, choose 3 memes out of 5: C(5,3) = 10.Then, assign these 3 memes to A, B, and C: 3! = 6 ways.Then, assign the remaining 2 memes to the remaining 7 colleagues: P(7,2) = 7 * 6 = 42.So, total number of favorable distributions is 10 * 6 * 42.Compute that:10 * 6 = 6060 * 42 = 2,520So, number of favorable distributions is 2,520.Therefore, the probability is 2,520 / 30,240.Simplify that:Divide numerator and denominator by 2,520:2,520 / 2,520 = 130,240 / 2,520 = 12So, probability is 1/12.Wait, is that correct? Let me double-check.Alternatively, another approach:The total number of ways to distribute 5 distinct memes to 10 colleagues is P(10,5) = 30,240.The number of favorable ways:- We need A, B, and C each to get exactly one meme, and the other two memes to go to the remaining 7.So, first, assign one meme to A: 5 choices.Then, assign one meme to B: 4 remaining choices.Then, assign one meme to C: 3 remaining choices.Then, assign the remaining 2 memes to the remaining 7 colleagues: P(7,2) = 7 * 6 = 42.So, total number of favorable ways is 5 * 4 * 3 * 42.Compute that:5 * 4 = 2020 * 3 = 6060 * 42 = 2,520Same as before.So, probability is 2,520 / 30,240 = 1/12.Yes, that seems correct.Alternatively, another way to think about it is:The probability that A gets a meme is 5/10.Then, given A has a meme, the probability that B gets a meme is 4/9.Then, given A and B have memes, the probability that C gets a meme is 3/8.Then, the remaining two memes need to go to the remaining 7 colleagues.But wait, actually, the probability that exactly A, B, and C get one meme each, and the other two go to the remaining 7.But this approach might be more complicated because we have to consider the exact distribution.Alternatively, using combinations:The number of ways to choose 3 specific people (A, B, C) to receive one meme each, and the other two to receive from the remaining 7.So, the number of favorable distributions is:C(5,3) * 3! * P(7,2) = 10 * 6 * 42 = 2,520.Which is the same as before.So, probability is 2,520 / 30,240 = 1/12.Yes, that seems correct.So, Sub-problem 2 answer is 1/12.But let me think again to make sure.Another approach: The probability that exactly A, B, and C each get one meme, and the other two go to the remaining 7.So, the total number of ways to distribute 5 memes to 10 people is 10P5 = 30,240.The number of favorable ways:- Choose 3 memes for A, B, C: C(5,3) = 10.- Assign these 3 memes to A, B, C: 3! = 6.- Assign the remaining 2 memes to any of the remaining 7: P(7,2) = 42.Total: 10 * 6 * 42 = 2,520.Probability: 2,520 / 30,240 = 1/12.Yes, that's consistent.Alternatively, think of it as:The probability that A gets a meme is 5/10.Then, given A has a meme, the probability that B gets a meme is 4/9.Then, given A and B have memes, the probability that C gets a meme is 3/8.Then, the remaining two memes must go to the remaining 7 colleagues, which is P(7,2) / P(7,2) = 1, but actually, we have to consider the total number of ways.Wait, maybe this approach is not as straightforward.Alternatively, using hypergeometric distribution.But I think the first method is correct.So, in conclusion, Sub-problem 1 answer is 61,124,064, and Sub-problem 2 answer is 1/12.But wait, let me check Sub-problem 1 again because I'm a bit uncertain.If the problem allows reusing images and captions, then it's 96C5. If not, it's 5,322,240.But the problem says "create a set of 5 distinct memes, each meme consisting of one image and one caption." It doesn't specify that images or captions can't be reused. So, I think it's 96C5.But let me think about it in another way. If each meme is a unique combination, then the total number is 12*8=96, and choosing 5 is 96C5.Yes, that makes sense.So, I think Sub-problem 1 is 61,124,064.Sub-problem 2 is 1/12.Wait, but 61,124,064 is a huge number. Maybe I should write it in terms of factorials or something else, but I think it's fine.Alternatively, maybe the problem expects the second interpretation, where images and captions can't be reused, so the answer is 5,322,240.But without more context, it's hard to say. But given the problem statement, I think the first interpretation is correct.So, final answers:Sub-problem 1: 61,124,064 ways.Sub-problem 2: Probability is 1/12.</think>

🧠:<think>Okay, so I need to tackle this problem about the SIR model. Let me start by understanding what the SIR model is. From what I remember, SIR stands for Susceptible, Infected, Recovered. It's a way to model the spread of infectious diseases in a population. The model divides the population into three compartments: people who are susceptible to the disease, those who are currently infected, and those who have recovered and are now immune.The problem gives me specific parameters for a country in the European Union. The total population N is 10 million. The initial number of infected individuals I₀ is 500, and the initial number of recovered R₀ is 0. The infection rate β is 0.3, and the recovery rate γ is 0.1.First, I need to write down the system of differential equations for the SIR model. I think the standard SIR model has the following equations:dS/dt = -β * S * I / NdI/dt = β * S * I / N - γ * IdR/dt = γ * ILet me verify that. Yes, the susceptible population decreases when they come into contact with infected individuals, which is why it's negative beta times S times I over N. The infected population increases from susceptibles getting infected but decreases as people recover, which is gamma times I. The recovered population increases as infected individuals recover.So, plugging in the given parameters, N is 10 million, which is 10,000,000. So, the equations become:dS/dt = -0.3 * S * I / 10,000,000dI/dt = 0.3 * S * I / 10,000,000 - 0.1 * IdR/dt = 0.1 * IBut wait, actually, since N is 10 million, and S + I + R = N, we can write S(t) = N - I(t) - R(t). But for the differential equations, we can keep them as is.Now, the first part is to write these equations, which I think I have done. The next step is to solve these differential equations to find S(t), I(t), and R(t). Hmm, solving the SIR model analytically is tricky because it's a nonlinear system due to the term S*I. I remember that the SIR model doesn't have a closed-form solution in general, so we usually solve it numerically. But maybe there's a way to express it in terms of integrals or something?Wait, let me think. For the SIR model, sometimes we can use the concept of the basic reproduction number, R0, which is β / γ. In this case, R0 would be 0.3 / 0.1 = 3. So, R0 is 3, which is greater than 1, meaning the disease will spread.But solving the differential equations analytically... I think it's not straightforward. Maybe I can use some substitution or integrate factors? Let me try.Looking at the equations:dS/dt = -β S I / NdI/dt = β S I / N - γ IdR/dt = γ ISince dS/dt + dI/dt + dR/dt = 0, the total population remains constant.I know that in the SIR model, sometimes we can express dI/dt in terms of S. Let's see.From dS/dt = -β S I / N, we can write dS = -β S I / N dt.But maybe it's better to consider the ratio dI/dS. Let me try that.From the two equations:dI/dt = β S I / N - γ IdS/dt = -β S I / NSo, dI/dS = (dI/dt) / (dS/dt) = [β S I / N - γ I] / (-β S I / N) = [β S I / N - γ I] / (-β S I / N)Simplify numerator:β S I / N - γ I = I (β S / N - γ)Denominator: -β S I / NSo, dI/dS = [I (β S / N - γ)] / (-β S I / N) = (β S / N - γ) / (-β S / N) = (- (β S / N - γ)) / (β S / N) = (-β S / N + γ) / (β S / N) = (-β S / N)/(β S / N) + γ / (β S / N) = -1 + (γ N)/(β S)So, dI/dS = -1 + (γ N)/(β S)That's a separable equation. Let me write it as:dI/dS = -1 + (γ N)/(β S)Which can be rewritten as:dI = [ -1 + (γ N)/(β S) ] dSIntegrate both sides:∫ dI = ∫ [ -1 + (γ N)/(β S) ] dSSo,I = -S + (γ N)/(β) ln S + CWhere C is the constant of integration.Now, we can use the initial conditions to find C. At t=0, S(0) = N - I₀ - R₀ = 10,000,000 - 500 - 0 = 9,999,500. I(0) = 500.So, plugging into the equation:500 = -9,999,500 + (γ N)/(β) ln(9,999,500) + CCompute (γ N)/(β): γ is 0.1, N is 10,000,000, β is 0.3.So, (0.1 * 10,000,000)/0.3 = 1,000,000 / 0.3 ≈ 3,333,333.333So,500 = -9,999,500 + 3,333,333.333 * ln(9,999,500) + CCompute ln(9,999,500). Since 9,999,500 is approximately 10,000,000, ln(10,000,000) is ln(10^7) = 7 ln(10) ≈ 7 * 2.302585 ≈ 16.1181But 9,999,500 is slightly less than 10,000,000. Let me compute ln(9,999,500).Let me approximate it. Let x = 10,000,000, so ln(x - 500) ≈ ln(x) - 500/x. Because ln(x - Δx) ≈ ln(x) - Δx/x for small Δx.So, ln(9,999,500) ≈ ln(10,000,000) - 500 / 10,000,000 ≈ 16.1181 - 0.00005 ≈ 16.11805So,500 ≈ -9,999,500 + 3,333,333.333 * 16.11805 + CCompute 3,333,333.333 * 16.11805:First, 3,333,333.333 * 16 = 53,333,333.328Then, 3,333,333.333 * 0.11805 ≈ 3,333,333.333 * 0.1 = 333,333.3333Plus 3,333,333.333 * 0.01805 ≈ 3,333,333.333 * 0.01 = 33,333.3333Plus 3,333,333.333 * 0.00805 ≈ 26,851.666So total ≈ 333,333.3333 + 33,333.3333 + 26,851.666 ≈ 393,518.3326So total ≈ 53,333,333.328 + 393,518.3326 ≈ 53,726,851.66So,500 ≈ -9,999,500 + 53,726,851.66 + CCompute -9,999,500 + 53,726,851.66 ≈ 43,727,351.66So,500 ≈ 43,727,351.66 + CThus, C ≈ 500 - 43,727,351.66 ≈ -43,726,851.66So, the equation becomes:I = -S + (γ N)/(β) ln S + C ≈ -S + 3,333,333.333 ln S - 43,726,851.66But this seems a bit messy. Maybe I made a mistake in the integration or the approximation.Wait, perhaps I should have used the exact value of ln(9,999,500). Let me compute it more accurately.Using a calculator, ln(9,999,500) ≈ ln(10^7 - 500). Let me compute it numerically.But since I don't have a calculator here, maybe I can use the Taylor series expansion around x=10^7.Let f(x) = ln(x). Then f(x - Δx) ≈ f(x) - Δx / x - (Δx)^2 / (2x^2) - ...So, f(10^7 - 500) ≈ ln(10^7) - 500 / 10^7 - (500)^2 / (2*(10^7)^2)Compute:ln(10^7) ≈ 16.118092500 / 10^7 = 0.00005(500)^2 / (2*(10^7)^2) = 250,000 / (2*10^14) = 125,000 / 10^14 = 1.25e-9So, ln(9,999,500) ≈ 16.118092 - 0.00005 - 0.00000000125 ≈ 16.11804199875So, more accurately, it's approximately 16.118042.So, going back:500 = -9,999,500 + 3,333,333.333 * 16.118042 + CCompute 3,333,333.333 * 16.118042:First, 3,333,333.333 * 16 = 53,333,333.328Then, 3,333,333.333 * 0.118042 ≈Compute 3,333,333.333 * 0.1 = 333,333.33333,333,333.333 * 0.018042 ≈ 3,333,333.333 * 0.01 = 33,333.33333,333,333.333 * 0.008042 ≈ 26,807.1428So total ≈ 333,333.3333 + 33,333.3333 + 26,807.1428 ≈ 393,473.8094So total ≈ 53,333,333.328 + 393,473.8094 ≈ 53,726,807.137So,500 ≈ -9,999,500 + 53,726,807.137 + CCompute -9,999,500 + 53,726,807.137 ≈ 43,727,307.137So,500 ≈ 43,727,307.137 + CThus, C ≈ 500 - 43,727,307.137 ≈ -43,726,807.137So, the equation is:I = -S + 3,333,333.333 ln S - 43,726,807.137This is an implicit equation relating I and S. To find S(t), I(t), and R(t), we would need to solve this equation numerically, perhaps using methods like the Runge-Kutta method or Euler's method.But since the problem asks for explicit expressions, I might need to reconsider. Maybe there's a way to express the solution in terms of integrals or use the concept of the final size of the epidemic.Wait, another approach is to use the fact that the SIR model can be analyzed using the final size equation, which gives the total number of people who have been infected by the end of the epidemic.The final size equation is given by:ln(S(∞)/S₀) = -R₀ (1 - S(∞)/N)Where S₀ is the initial susceptible population, which is 9,999,500.But I'm not sure if this helps me find S(t), I(t), and R(t) explicitly. It just gives the final size.Alternatively, maybe I can express the solution in terms of the inverse of the function involving the integral of 1/(S - S_c), where S_c is the critical threshold.Wait, let me recall. The SIR model can be analyzed by considering the threshold S_c = γ N / β. If S(0) > S_c, the epidemic will occur; otherwise, it will die out.In this case, S_c = (0.1 * 10,000,000)/0.3 ≈ 3,333,333.333. Since S(0) = 9,999,500 > S_c, the epidemic will occur.But I still need to find S(t), I(t), and R(t). Since the equations are nonlinear, I think the best approach is to solve them numerically. However, the problem asks for explicit expressions, which suggests that maybe I can express the solution in terms of integrals or use some substitution.Wait, another idea: sometimes the SIR model can be transformed into a system that can be integrated. Let me try to write the equations in terms of dimensionless variables.Let me define s = S/N, i = I/N, r = R/N. Then, s + i + r = 1.The equations become:ds/dt = -β s idi/dt = β s i - γ idr/dt = γ iThis might simplify things a bit. Now, we can write:ds/dt = -β s idi/dt = (β s - γ) idr/dt = γ iWe can also note that dr/dt = γ i, so integrating dr = γ i dt, but since we have di/dt in terms of i and s, maybe we can find a relationship between s and i.From ds/dt = -β s i and di/dt = (β s - γ) i, we can write:di/ds = (β s - γ) i / (-β s i) = (β s - γ)/(-β s) = (-β s + γ)/β s = -1 + γ/(β s)Which is the same as before. So, we end up with the same equation:di/ds = -1 + (γ / β) (1/s)Which is what I had earlier. So, integrating this gives:i = -s + (γ / β) ln s + CBut since s = S/N, and S = N s, we can write:I = N i = N (-s + (γ / β) ln s + C) = -N s + (γ N / β) ln s + C'Where C' = N C.But we already did this earlier. So, perhaps I need to accept that the solution can't be expressed in a simple closed-form and instead present the implicit solution or refer to numerical methods.Alternatively, maybe I can express the solution in terms of the inverse function. For example, express t as a function of S.From dS/dt = -β S I / N, and I can be expressed in terms of S from the implicit equation.So, let's try that.From the implicit equation:I = -S + (γ N / β) ln(S) + CBut we have C ≈ -43,726,807.137, as computed earlier.So,I = -S + (γ N / β) ln(S) + CBut since I = N i and S = N s, we can write:i = -s + (γ / β) ln(s) + C/NBut perhaps it's better to keep it in terms of S.So, dS/dt = -β S I / N = -β S / N * [ -S + (γ N / β) ln(S) + C ]This is a separable equation:dS / [ -S + (γ N / β) ln(S) + C ] = -β / N dtIntegrate both sides:∫ [1 / ( -S + (γ N / β) ln(S) + C ) ] dS = -β / N ∫ dtBut this integral is not elementary, meaning it can't be expressed in terms of elementary functions. So, we can't solve for S(t) explicitly in terms of elementary functions. Therefore, the solution must be expressed implicitly or solved numerically.Given that, I think the answer to part 1 is that the system of differential equations is as I wrote earlier, and the solution can be expressed implicitly as:I = -S + (γ N / β) ln(S) + Cwith C determined by the initial conditions, but explicit expressions for S(t), I(t), and R(t) require numerical methods.Alternatively, perhaps the problem expects me to set up the equations and recognize that an explicit solution isn't feasible, so I can describe the method to solve them numerically.But the problem says "solve these equations to find the number of susceptible (S(t)), infected (I(t)), and recovered (R(t)) individuals as a function of time (t)." So, maybe I need to present the implicit solution as the answer.Alternatively, perhaps I can express the solution in terms of the inverse function, but I don't think that's necessary.Wait, maybe I can use the fact that the SIR model can be solved using the Lambert W function in some cases, but I'm not sure if that applies here.Alternatively, perhaps I can express the solution in terms of the integral:t = ∫ [1 / (β S I / N - γ I)] dIBut that seems complicated.Wait, let's try to express t as a function of I.From dI/dt = β S I / N - γ IBut S = N - I - R, and R = γ ∫ I dt from 0 to t.But that introduces an integral, making it difficult.Alternatively, since dR/dt = γ I, we can write R(t) = γ ∫₀ᵗ I(τ) dτ.But again, without knowing I(t), we can't proceed.So, I think the conclusion is that the SIR model equations are as written, and the solution requires numerical methods to find S(t), I(t), and R(t). Therefore, the explicit expressions aren't possible in terms of elementary functions, but the implicit relationship is as derived.Moving on to part 2: Linearize the system around the disease-free equilibrium and determine the eigenvalues of the Jacobian matrix.First, what is the disease-free equilibrium? It's when I = 0 and R = 0, so S = N. So, the equilibrium point is (S, I, R) = (N, 0, 0).To linearize the system around this equilibrium, we need to compute the Jacobian matrix of the system evaluated at (N, 0, 0).The Jacobian matrix J is given by:[ ∂(dS/dt)/∂S , ∂(dS/dt)/∂I , ∂(dS/dt)/∂R ][ ∂(dI/dt)/∂S , ∂(dI/dt)/∂I , ∂(dI/dt)/∂R ][ ∂(dR/dt)/∂S , ∂(dR/dt)/∂I , ∂(dR/dt)/∂R ]Compute each partial derivative.First, dS/dt = -β S I / NSo,∂(dS/dt)/∂S = -β I / N∂(dS/dt)/∂I = -β S / N∂(dS/dt)/∂R = 0Next, dI/dt = β S I / N - γ ISo,∂(dI/dt)/∂S = β I / N∂(dI/dt)/∂I = β S / N - γ∂(dI/dt)/∂R = 0Finally, dR/dt = γ ISo,∂(dR/dt)/∂S = 0∂(dR/dt)/∂I = γ∂(dR/dt)/∂R = 0Now, evaluate the Jacobian at (N, 0, 0):For ∂(dS/dt)/∂S: -β * 0 / N = 0∂(dS/dt)/∂I: -β * N / N = -β∂(dS/dt)/∂R: 0For ∂(dI/dt)/∂S: β * 0 / N = 0∂(dI/dt)/∂I: β * N / N - γ = β - γ∂(dI/dt)/∂R: 0For ∂(dR/dt)/∂S: 0∂(dR/dt)/∂I: γ∂(dR/dt)/∂R: 0So, the Jacobian matrix J at (N, 0, 0) is:[ 0 , -β , 0 ][ 0 , β - γ , 0 ][ 0 , γ , 0 ]Now, to find the eigenvalues, we solve det(J - λ I) = 0.The matrix J - λ I is:[ -λ , -β , 0 ][ 0 , β - γ - λ , 0 ][ 0 , γ , -λ ]The determinant is:-λ * ( (β - γ - λ)(-λ) - 0 ) - (-β) * (0 - 0) + 0 * ... = -λ * ( -λ (β - γ - λ) ) = -λ * (-λ β + λ γ + λ²) = λ² (β - γ) - λ³Wait, let me compute it step by step.The determinant of a 3x3 matrix:| a b c || d e f || g h i |is a(ei - fh) - b(di - fg) + c(dh - eg)So, for our matrix:a = -λ, b = -β, c = 0d = 0, e = β - γ - λ, f = 0g = 0, h = γ, i = -λSo,det = (-λ)[(β - γ - λ)(-λ) - 0] - (-β)[0*(-λ) - 0*0] + 0[0*γ - (β - γ - λ)*0]Simplify:= (-λ)[ -λ (β - γ - λ) ] - (-β)(0) + 0= (-λ)( -λ β + λ γ + λ² )= λ² (β - γ) - λ³So, the characteristic equation is:-λ³ + λ² (β - γ) = 0Factor:λ² (-λ + β - γ) = 0So, the eigenvalues are λ = 0 (double root) and λ = β - γ.Given our parameters, β = 0.3, γ = 0.1, so λ = 0.3 - 0.1 = 0.2.So, the eigenvalues are 0, 0, and 0.2.Wait, but that can't be right because the Jacobian is a 3x3 matrix, so it should have 3 eigenvalues. But in this case, two of them are zero and one is 0.2.But wait, looking back at the Jacobian matrix:[ 0 , -β , 0 ][ 0 , β - γ , 0 ][ 0 , γ , 0 ]This matrix has a block structure. The first row and column form a 1x1 block with eigenvalue 0. The remaining 2x2 block is:[ β - γ , 0 ][ γ , 0 ]Wait, no, actually, the Jacobian is:Row 1: [0, -β, 0]Row 2: [0, β - γ, 0]Row 3: [0, γ, 0]So, it's a lower triangular matrix? No, because the third row has a non-zero element in the second column.Wait, actually, the matrix is:Column 1: [0, 0, 0]Column 2: [-β, β - γ, γ]Column 3: [0, 0, 0]So, it's a rank 1 matrix because only the second column has non-zero elements. Therefore, the eigenvalues are 0 (with multiplicity 2) and the trace of the matrix, which is 0 + (β - γ) + 0 = β - γ. So, the eigenvalues are 0, 0, and β - γ.Yes, that makes sense. So, the eigenvalues are 0 (with multiplicity 2) and β - γ.Given β = 0.3 and γ = 0.1, β - γ = 0.2, which is positive. Therefore, one eigenvalue is positive, and the other two are zero.In terms of stability, for a disease-free equilibrium, if all eigenvalues have negative real parts, the equilibrium is stable. If any eigenvalue has a positive real part, it's unstable.Here, one eigenvalue is positive (0.2), so the disease-free equilibrium is unstable. This means that if there's a small perturbation (like introducing a few infected individuals), the disease will spread, which aligns with our earlier calculation of R0 = 3 > 1.So, the disease-free equilibrium is unstable because one of the eigenvalues is positive.Therefore, the analysis shows that the disease-free equilibrium is unstable, and the disease will spread in the population.Final Answer1. The system of differential equations is: [ frac{dS}{dt} = -0.3 cdot frac{S cdot I}{10,000,000} ] [ frac{dI}{dt} = 0.3 cdot frac{S cdot I}{10,000,000} - 0.1 cdot I ] [ frac{dR}{dt} = 0.1 cdot I ] The solution can be expressed implicitly as: [ I = -S + frac{gamma N}{beta} ln(S) + C ] where ( C ) is determined by initial conditions. Explicit solutions require numerical methods.2. The eigenvalues of the Jacobian matrix at the disease-free equilibrium are ( 0 ) (with multiplicity 2) and ( 0.2 ). Since one eigenvalue is positive, the disease-free equilibrium is unstable.The final answers are:1. The differential equations are as above, and the solution requires numerical methods.2. The eigenvalues are (boxed{0}) (double root) and (boxed{0.2}), indicating the disease-free equilibrium is unstable.</think>

🧠:<think>Alright, so I have two investment problems to solve. Let me tackle them one by one. Starting with Investment A. It's about continuous compounding interest. I remember that continuous compounding uses the formula involving e, the base of natural logarithms. The future value F(t) should be P multiplied by e raised to the power of rt, right? So, F(t) = P * e^(rt). Okay, so if I need to find the future value after t years, that's straightforward. But the second part asks me to find the required interest rate r if the future value needs to be 500,000 in 10 years with an initial investment of 100,000. Let me write down the formula again: F(t) = P * e^(rt)We know F(t) is 500,000, P is 100,000, and t is 10 years. So plugging in the numbers:500,000 = 100,000 * e^(10r)To solve for r, I can divide both sides by 100,000:5 = e^(10r)Now, take the natural logarithm of both sides to get rid of the exponential:ln(5) = 10rSo, r = ln(5)/10Calculating that, ln(5) is approximately 1.6094, so dividing by 10 gives r ≈ 0.16094, or 16.094%. That seems pretty high, but continuous compounding does grow faster, so maybe that's correct.Moving on to Investment B. This one is more complex because it involves stochastic differential equations. The SDE given is dX(t) = μ X(t) dt + σ X(t) dW(t). This looks like a geometric Brownian motion model, which is commonly used in finance for stock prices.The finance enthusiast wants to maximize the expected utility of their terminal wealth. The utility function is given as U(W) = (W^(1-γ))/(1-γ), where γ is the relative risk aversion. I need to find the optimal expected return μ* that maximizes this expected utility.First, I recall that for a utility function of the form U(W) = W^(1-γ)/(1-γ), the optimal portfolio problem can be solved using the concept of expected utility maximization under the geometric Brownian motion model.In such cases, the optimal strategy involves choosing a portfolio that balances the expected return and the risk (volatility). The optimal μ* can be derived using the Hamilton-Jacobi-Bellman (HJB) equation, but maybe there's a simpler way since the utility function is a power function.Alternatively, I remember that for power utility functions, the optimal portfolio is given by μ* = (μ - (σ^2 γ)/2) / (1 - γ). Wait, no, that might be the optimal drift in the context of the SDE.Wait, let me think again. The problem is about maximizing the expected utility E[U(X(T))]. Since the utility function is U(W) = W^(1-γ)/(1-γ), the expected utility is E[X(T)^(1-γ)/(1-γ)].Given that X(t) follows a geometric Brownian motion, the solution to the SDE is X(T) = X(0) * exp[(μ - σ^2/2)T + σ W(T)]. So, the expected utility becomes E[(X(0) * exp[(μ - σ^2/2)T + σ W(T)])^(1-γ)/(1-γ)].Simplifying, since X(0) is 200,000 and T is 5 years, we can write:E[U(X(T))] = E[(200,000 * exp[(μ - σ^2/2)*5 + σ W(5)])^(1-γ)/(1-γ)]Let me denote Y = exp[(μ - σ^2/2)*5 + σ W(5)]. Then, X(T) = 200,000 * Y, so U(X(T)) = (200,000 * Y)^(1-γ)/(1-γ).Therefore, E[U(X(T))] = (200,000^(1-γ)/(1-γ)) * E[Y^(1-γ)].Now, since Y is lognormally distributed, E[Y^k] = exp[(k(μ - σ^2/2) + (k^2 σ^2)/2) * T]. In our case, k = 1 - γ, and T = 5. So,E[Y^(1-γ)] = exp[( (1 - γ)(μ - σ^2/2) + ((1 - γ)^2 σ^2)/2 ) * 5 ]Therefore, the expected utility is:(200,000^(1-γ)/(1-γ)) * exp[ ( (1 - γ)(μ - σ^2/2) + ( (1 - γ)^2 σ^2 ) / 2 ) * 5 ]To maximize this with respect to μ, we can take the derivative with respect to μ and set it to zero. However, looking at the expression, the only term involving μ is inside the exponent, specifically:( (1 - γ)(μ - σ^2/2) + ... )So, the exponent is linear in μ. Therefore, the expected utility is maximized when μ is as large as possible. But wait, that doesn't make sense because μ is the expected return, which is a parameter of the SDE. In reality, μ is not something we can choose arbitrarily; it's a parameter of the model. Wait, perhaps I misunderstood the problem. It says "find the optimal expected return μ* that maximizes the expected utility." So, maybe μ is a parameter we can choose, perhaps by selecting a portfolio with a certain expected return. But in the SDE, μ is the drift, which is typically a parameter that can be chosen based on the investor's risk preferences.In the context of Merton's portfolio problem, the optimal portfolio choice involves selecting a proportion of wealth to invest in the risky asset (with drift μ and volatility σ) and the rest in the risk-free asset (with drift r). However, in this problem, it's just a single risky asset, and the investor can choose the proportion to invest. But the problem states the SDE as dX(t) = μ X(t) dt + σ X(t) dW(t), so perhaps μ is the expected return of the portfolio, which can be adjusted by the investor.Assuming that μ can be chosen, perhaps through allocation between a risky asset and a risk-free asset, then the optimal μ* would be determined by the investor's risk aversion.Wait, in the standard Merton problem, the optimal portfolio is to invest a fraction of wealth in the risky asset such that the expected return of the portfolio is μ_p = r + (μ - r) * (1 - γ) * (σ^2)^{-1} * (something). Hmm, maybe I need to recall the formula.Alternatively, perhaps using the concept of the certainty equivalent. The expected utility is maximized when the investor chooses the portfolio that maximizes E[U(X(T))]. Given that, and knowing that the expected utility expression is:E[U(X(T))] = (200,000^(1-γ)/(1-γ)) * exp[ ( (1 - γ)(μ - σ^2/2) + ( (1 - γ)^2 σ^2 ) / 2 ) * 5 ]Simplify the exponent:Let me compute the exponent:= (1 - γ)(μ - σ^2/2) * 5 + ( (1 - γ)^2 σ^2 ) / 2 * 5= 5(1 - γ)μ - 5(1 - γ)σ^2/2 + 5(1 - γ)^2 σ^2 / 2Combine the terms:= 5(1 - γ)μ + [ -5(1 - γ)σ^2/2 + 5(1 - γ)^2 σ^2 / 2 ]Factor out 5(1 - γ)σ^2 / 2:= 5(1 - γ)μ + 5(1 - γ)σ^2 / 2 [ -1 + (1 - γ) ]Simplify inside the brackets:-1 + 1 - γ = -γSo,= 5(1 - γ)μ - 5(1 - γ)γ σ^2 / 2Therefore, the exponent simplifies to:5(1 - γ)μ - (5/2)(1 - γ)γ σ^2Thus, the expected utility becomes:(200,000^(1-γ)/(1-γ)) * exp[5(1 - γ)μ - (5/2)(1 - γ)γ σ^2]To maximize this with respect to μ, we can take the derivative with respect to μ and set it to zero. However, looking at the exponent, it's linear in μ. The term involving μ is 5(1 - γ)μ. Since 1 - γ is positive (because γ > 0, but less than 1 for risk aversion), the exponent increases as μ increases. Therefore, to maximize the expected utility, we should set μ as large as possible. But that can't be right because in reality, μ is bounded by the market conditions.Wait, perhaps I made a mistake. Maybe μ is not a variable we can choose, but rather, the investor can choose the proportion of wealth to invest in the risky asset, which affects the overall μ of the portfolio. Let me think again. Suppose the investor can choose to invest a fraction π(t) of their wealth in the risky asset, which has drift μ and volatility σ, and the rest in a risk-free asset with rate r. Then, the portfolio's drift would be μ_p = π μ + (1 - π) r, and the volatility would be σ_p = π σ.But in the given SDE, it's just dX(t) = μ X(t) dt + σ X(t) dW(t). So maybe μ is the expected return of the portfolio, which can be adjusted by choosing π. Therefore, the investor can choose μ by selecting π.Assuming that, then the optimal μ* would be determined by the investor's risk aversion. In the Merton problem, the optimal proportion π* is given by:π* = (μ - r)/(σ^2 γ)But in our case, there's no mention of a risk-free rate. Maybe we can assume that the risk-free rate is zero, or perhaps it's incorporated into μ. Alternatively, maybe the problem is considering only the risky asset, and μ is the expected return that can be adjusted by the investor's choice of portfolio.Wait, the problem says "find the optimal expected return μ* that maximizes the expected utility." So, perhaps μ is a variable we can choose, and we need to find the μ that maximizes the expected utility.Given that, and from the earlier expression, the expected utility is:E[U] = C * exp[5(1 - γ)μ - (5/2)(1 - γ)γ σ^2]Where C is a constant (200,000^(1-γ)/(1-γ)). To maximize E[U], we need to maximize the exponent with respect to μ. The exponent is linear in μ: 5(1 - γ)μ - constant. Since 1 - γ is positive (assuming γ < 1, which is typical for risk aversion), the exponent increases as μ increases. Therefore, to maximize E[U], we should set μ as large as possible. But that's not practical because μ can't be infinite.Wait, perhaps I'm missing something. Maybe μ is not a variable we can choose, but rather, it's given, and the investor can choose the proportion π to get a portfolio drift μ_p. Alternatively, perhaps the problem assumes that μ can be chosen freely, and we need to find the optimal μ that maximizes the expected utility, given the utility function. In that case, since the exponent is 5(1 - γ)μ - (5/2)(1 - γ)γ σ^2, and we can choose μ, then to maximize the exponent, we set μ as large as possible. But that's not feasible because μ is bounded by the market. Alternatively, maybe the problem is considering that the investor can choose μ by selecting a portfolio, and the optimal μ is determined by the risk aversion parameter γ. Wait, perhaps I should use the concept of the certainty equivalent. The certainty equivalent is the amount of wealth that provides the same utility as the expected utility. But I'm getting confused. Let me try a different approach. The expected utility is E[U(X(T))] = E[ (X(T)^(1-γ))/(1-γ) ]Given that X(T) follows a lognormal distribution, the expected utility can be written as:E[U(X(T))] = (X(0)^(1-γ)/(1-γ)) * E[ exp( (1 - γ)(μ - σ^2/2)T + (1 - γ)σ W(T) ) ]But since W(T) is a normal variable with mean 0 and variance T, the expectation of exp(a + b W(T)) is exp(a + b^2 T / 2). So, let me compute E[ exp( (1 - γ)(μ - σ^2/2)T + (1 - γ)σ W(T) ) ]Let a = (1 - γ)(μ - σ^2/2)T and b = (1 - γ)σ sqrt(T). Wait, no, actually, W(T) has variance T, so the expectation is:E[exp(a + b W(T))] = exp(a + b^2 / 2)But in our case, the exponent is (1 - γ)(μ - σ^2/2)T + (1 - γ)σ W(T). So, a = (1 - γ)(μ - σ^2/2)T and b = (1 - γ)σ.Therefore, E[exp(a + b W(T))] = exp(a + b^2 / 2) = exp[ (1 - γ)(μ - σ^2/2)T + ( (1 - γ)^2 σ^2 ) / 2 ]Thus, the expected utility is:(200,000^(1-γ)/(1-γ)) * exp[ (1 - γ)(μ - σ^2/2)*5 + ( (1 - γ)^2 σ^2 ) / 2 ]Simplify the exponent:= 5(1 - γ)(μ - σ^2/2) + ( (1 - γ)^2 σ^2 ) / 2= 5(1 - γ)μ - 5(1 - γ)σ^2/2 + (1 - γ)^2 σ^2 / 2Factor out (1 - γ)σ^2 / 2:= 5(1 - γ)μ + (1 - γ)σ^2 / 2 [ -5 + (1 - γ) ]Simplify inside the brackets:-5 + 1 - γ = -4 - γWait, that doesn't seem right. Let me recalculate:Wait, the terms are:5(1 - γ)μ - (5/2)(1 - γ)σ^2 + (1/2)(1 - γ)^2 σ^2Factor out (1 - γ)σ^2 / 2:= 5(1 - γ)μ + (1 - γ)σ^2 / 2 [ -5 + (1 - γ) ]Yes, that's correct.So, inside the brackets: -5 + 1 - γ = -4 - γThus, the exponent becomes:5(1 - γ)μ - (1 - γ)σ^2 / 2 * (4 + γ)Wait, no, because it's (1 - γ)σ^2 / 2 multiplied by (-5 + 1 - γ) which is (-4 - γ). So,= 5(1 - γ)μ - (1 - γ)σ^2 / 2 * (4 + γ)But this seems complicated. Maybe I should keep it as:Exponent = 5(1 - γ)μ - (5/2)(1 - γ)σ^2 + (1/2)(1 - γ)^2 σ^2Now, to maximize the expected utility with respect to μ, we take the derivative of the exponent with respect to μ and set it to zero.The exponent is linear in μ, so the derivative is 5(1 - γ). Setting this equal to zero would imply 5(1 - γ) = 0, which would require γ = 1. But γ > 0, and typically γ ≠ 1 because the utility function becomes logarithmic when γ approaches 1. This suggests that the expected utility is increasing in μ, so to maximize it, we should set μ as large as possible. However, in reality, μ can't be increased indefinitely because it's bounded by the market's expected returns. Wait, perhaps I'm misunderstanding the problem. Maybe μ is not a variable we can choose, but rather, the investor can choose the proportion of wealth invested in the risky asset, which affects the overall μ of the portfolio. In that case, let's denote π as the proportion invested in the risky asset. Then, the portfolio's drift μ_p = π μ + (1 - π) r, where r is the risk-free rate. However, the problem doesn't mention a risk-free rate, so perhaps we can assume r = 0 or it's incorporated into μ. Alternatively, if we consider that the investor can choose π to adjust μ_p, then the optimal π would be chosen to maximize the expected utility. In the standard Merton problem, the optimal π is given by:π* = (μ - r)/(σ^2 γ)Assuming r = 0, then π* = μ/(σ^2 γ). But then μ_p = π* μ = μ^2/(σ^2 γ). But in our case, we need to find μ*, which is the optimal expected return. So, perhaps μ* is the optimal drift of the portfolio, which is μ_p = π* μ = (μ - r)/(σ^2 γ) * μ. Wait, this is getting too convoluted. Maybe I should recall that for a power utility function, the optimal portfolio is given by:π* = (μ - r)/(σ^2 γ)Assuming r = 0, then π* = μ/(σ^2 γ). But then, the portfolio's expected return is μ_p = π* μ = μ^2/(σ^2 γ). Wait, that doesn't seem right. Let me think again. If the investor invests a fraction π in the risky asset, then the portfolio's drift is μ_p = π μ + (1 - π) r. If r = 0, then μ_p = π μ. The volatility is σ_p = π σ. The expected utility is E[U(X(T))] = E[ (X(T)^(1-γ))/(1-γ) ]Given that X(T) follows a lognormal distribution with parameters μ_p T and σ_p^2 T, the expected utility can be written as:E[U] = (X(0)^(1-γ)/(1-γ)) * exp[ (1 - γ)(μ_p - σ_p^2 / 2) T + ( (1 - γ)^2 σ_p^2 T ) / 2 ]Substituting μ_p = π μ and σ_p = π σ:E[U] = (200,000^(1-γ)/(1-γ)) * exp[ (1 - γ)(π μ - (π^2 σ^2)/2 ) * 5 + ( (1 - γ)^2 π^2 σ^2 * 5 ) / 2 ]Simplify the exponent:= 5(1 - γ)π μ - (5/2)(1 - γ)π^2 σ^2 + (5/2)(1 - γ)^2 π^2 σ^2= 5(1 - γ)π μ + (5/2)(1 - γ)π^2 σ^2 [ -1 + (1 - γ) ]= 5(1 - γ)π μ + (5/2)(1 - γ)π^2 σ^2 ( -γ )= 5(1 - γ)π μ - (5/2)(1 - γ)γ π^2 σ^2To maximize this with respect to π, take the derivative with respect to π and set it to zero.d/dπ [5(1 - γ)π μ - (5/2)(1 - γ)γ π^2 σ^2] = 5(1 - γ)μ - 5(1 - γ)γ π σ^2 = 0Solving for π:5(1 - γ)μ = 5(1 - γ)γ π σ^2Cancel out 5(1 - γ):μ = γ π σ^2Thus, π = μ / (γ σ^2)Therefore, the optimal proportion π* is μ / (γ σ^2). But then, the optimal portfolio's expected return μ_p = π* μ = μ^2 / (γ σ^2)So, the optimal expected return μ* is μ^2 / (γ σ^2)Wait, but that seems a bit abstract. Maybe I should express μ* in terms of the given parameters. Wait, but in the problem, we are given X(0) = 200,000, time horizon T = 5, but we aren't given μ or σ. So, unless we have more information, we can't compute a numerical value for μ*. Wait, the problem says "find the optimal expected return μ* that maximizes the expected utility." So, perhaps μ* is expressed in terms of γ, σ, and possibly other given parameters. But looking back, the only given parameters are X(0) = 200,000 and T = 5. The utility function is given, but γ is just a parameter. So, unless we have specific values for γ and σ, we can't compute a numerical μ*. Wait, but maybe the problem assumes that μ is the expected return of the risky asset, and the investor can choose π to adjust the portfolio's expected return μ_p. Then, the optimal μ_p is given by the above expression. Alternatively, perhaps the problem is simpler. Since the utility function is U(W) = W^(1-γ)/(1-γ), the optimal portfolio is the one that maximizes the expected utility, which in the case of a single risky asset and a risk-free asset, leads to the optimal proportion π* = (μ - r)/(σ^2 γ). But since the problem doesn't mention a risk-free rate, maybe we can assume r = 0, so π* = μ/(σ^2 γ). Then, the optimal expected return μ_p = π* μ = μ^2/(σ^2 γ). But without knowing μ or σ, we can't find a numerical value for μ*. Wait, perhaps I'm overcomplicating. Maybe the optimal μ* is simply μ = (σ^2 γ)/2. No, that doesn't seem right. Alternatively, perhaps the optimal μ* is given by the certainty equivalent condition. The certainty equivalent is the amount C such that U(C) = E[U(X(T))]. For power utility, the certainty equivalent is C = E[X(T)]^(1-γ). But I'm not sure if that helps here. Wait, let me go back to the expected utility expression:E[U] = (200,000^(1-γ)/(1-γ)) * exp[5(1 - γ)μ - (5/2)(1 - γ)γ σ^2]To maximize this with respect to μ, since the exponent is linear in μ, the maximum occurs as μ approaches infinity, which is not practical. Therefore, perhaps the problem assumes that μ is a variable that can be chosen, but it's constrained by the market, meaning that μ cannot exceed a certain value. Alternatively, maybe the problem is asking for the optimal μ in terms of the risk aversion parameter γ and volatility σ, which would be μ* = (σ^2 γ)/2. Wait, let me think about the certainty equivalent. The certainty equivalent C satisfies:U(C) = E[U(X(T))]Which implies:C^(1-γ)/(1-γ) = E[X(T)^(1-γ)/(1-γ)]Therefore,C = [E[X(T)^(1-γ)]]^(1/(1-γ))But E[X(T)^(1-γ)] = X(0)^(1-γ) exp[ (1 - γ)(μ - σ^2/2)T + ( (1 - γ)^2 σ^2 T ) / 2 ]So,C = [200,000^(1-γ) exp(5(1 - γ)(μ - σ^2/2) + (5(1 - γ)^2 σ^2)/2 ) ]^(1/(1-γ))= 200,000 exp[ (5(1 - γ)(μ - σ^2/2) + (5(1 - γ)^2 σ^2)/2 ) / (1 - γ) ]Simplify the exponent:= 5(μ - σ^2/2) + (5(1 - γ) σ^2)/2= 5μ - (5 σ^2)/2 + (5 σ^2)/2 - (5 γ σ^2)/2= 5μ - (5 γ σ^2)/2Therefore, C = 200,000 exp(5μ - (5 γ σ^2)/2 )The certainty equivalent is a function of μ. To maximize C, we need to maximize the exponent 5μ - (5 γ σ^2)/2. Since this is linear in μ, C increases as μ increases. Therefore, again, to maximize C, we need to set μ as large as possible, which is not practical.This suggests that without constraints on μ, the optimal μ* is unbounded. But in reality, μ is limited by the market's expected returns. Therefore, perhaps the problem assumes that μ can be chosen freely, and the optimal μ* is the one that balances the expected return and risk according to the investor's risk aversion.Wait, maybe I should consider the problem differently. The investor wants to maximize E[U(X(T))], which is a function of μ. Since the exponent is linear in μ, the maximum occurs at the highest possible μ. But since μ is a parameter of the SDE, perhaps it's given, and the investor can't choose it. Alternatively, perhaps the problem is about choosing the drift μ of the portfolio by selecting the proportion π, as I thought earlier. In that case, the optimal μ_p is μ^2/(σ^2 γ), as derived earlier. But without knowing μ or σ, we can't compute a numerical value. Therefore, perhaps the answer is expressed in terms of γ and σ. Wait, the problem states "find the optimal expected return μ* that maximizes the expected utility." So, perhaps μ* is expressed as a function of γ and σ. From the earlier derivation, when we set the derivative to zero, we found that π = μ / (γ σ^2). Then, μ_p = π μ = μ^2 / (γ σ^2). But that's still in terms of μ, which is the expected return of the risky asset. If we assume that the risky asset's expected return μ is given, then μ_p is μ^2 / (γ σ^2). Alternatively, if we consider that the investor can choose μ by selecting π, then the optimal μ* is μ^2 / (γ σ^2). But without knowing μ or σ, we can't provide a numerical answer. Therefore, perhaps the problem expects an expression in terms of γ and σ. Alternatively, maybe I'm overcomplicating and the optimal μ* is simply μ = (σ^2 γ)/2. Wait, let me think about the utility maximization problem. For a power utility function, the optimal portfolio is the one that maximizes the expected utility, which involves balancing the expected return and the risk. The optimal μ* would be such that the marginal utility of wealth is proportional to the risk. In the case of a single risky asset, the optimal proportion π* is (μ - r)/(σ^2 γ). Assuming r = 0, π* = μ/(σ^2 γ). Therefore, the optimal expected return μ_p = π* μ = μ^2/(σ^2 γ). But again, without knowing μ or σ, we can't compute a numerical value. Wait, maybe the problem assumes that the investor can choose μ by selecting π, and the optimal μ* is given by μ* = (σ^2 γ)/2. Wait, let me think about the certainty equivalent again. The certainty equivalent C = 200,000 exp(5μ - (5 γ σ^2)/2 ). To maximize C, we set μ as large as possible. But if μ is constrained, say, by the market's expected return, then the optimal μ* would be the maximum possible μ. Alternatively, perhaps the problem is asking for the μ that makes the expected utility maximized, considering the risk aversion. Wait, maybe I should use the concept of the Arrow-Debreu utility function. For a power utility, the optimal portfolio is the one that maximizes the expected utility, which involves setting the portfolio's expected return such that the marginal utility of wealth is proportional to the risk. In the case of a single risky asset, the optimal μ* is given by μ* = r + (σ^2 γ)/2, where r is the risk-free rate. But since the problem doesn't mention a risk-free rate, perhaps r = 0, so μ* = (σ^2 γ)/2. But again, without knowing σ, we can't compute a numerical value. Wait, maybe the problem is simpler. Since the utility function is U(W) = W^(1-γ)/(1-γ), the optimal portfolio is the one that maximizes E[U(X(T))]. Given that X(T) follows a lognormal distribution, the expected utility is maximized when the portfolio's expected return is adjusted according to the investor's risk aversion. In the case of a single risky asset, the optimal expected return μ* is given by μ* = (σ^2 γ)/2. But I'm not entirely sure. Maybe I should look for a formula or derivation. Alternatively, perhaps the optimal μ* is given by μ* = (σ^2 γ)/2. Wait, let me think about the certainty equivalent again. The certainty equivalent C = 200,000 exp(5μ - (5 γ σ^2)/2 ). To maximize C, we need to maximize the exponent 5μ - (5 γ σ^2)/2. Since this is linear in μ, the maximum occurs as μ approaches infinity, which is not practical. Therefore, without constraints on μ, there's no finite optimal μ*. This suggests that the problem might have a typo or missing information, or perhaps I'm missing something. Alternatively, maybe the problem is asking for the optimal μ in terms of the certainty equivalent. Wait, perhaps the optimal μ* is such that the expected utility is maximized, which, given the exponent is linear in μ, implies that μ* is as large as possible. But since μ is a parameter of the SDE, perhaps it's given, and the investor can't choose it. Wait, maybe I'm overcomplicating. Let me try to summarize:For Investment A, the future value is F(t) = P e^(rt). To find r when F(10) = 500,000 and P = 100,000, we solve 500,000 = 100,000 e^(10r), which gives r = ln(5)/10 ≈ 16.09%.For Investment B, the optimal μ* is determined by maximizing the expected utility. Given the utility function and the SDE, the expected utility is maximized when μ is as large as possible, but since μ is a parameter, perhaps the optimal μ* is given by μ* = (σ^2 γ)/2. However, without knowing σ or γ, we can't compute a numerical value. Wait, but the problem doesn't provide values for γ or σ. It only gives X(0) = 200,000 and T = 5. Therefore, perhaps the answer is expressed in terms of γ and σ. Alternatively, maybe the optimal μ* is μ* = (σ^2 γ)/2. But I'm not entirely confident. Maybe I should look for a standard result. In the case of a power utility function, the optimal portfolio's expected return is given by μ_p = r + (σ^2 γ)/2, where r is the risk-free rate. If we assume r = 0, then μ_p = (σ^2 γ)/2. Therefore, the optimal expected return μ* is μ* = (σ^2 γ)/2. But since the problem doesn't provide σ or γ, perhaps the answer is expressed in terms of these parameters. Alternatively, if we consider that the investor can choose μ by selecting π, then μ* = μ^2/(σ^2 γ). But without knowing μ or σ, we can't compute a numerical value. Wait, perhaps the problem is asking for the optimal μ in terms of the given parameters, which are X(0) = 200,000 and T = 5. But since μ and σ aren't provided, maybe the answer is expressed as μ* = (σ^2 γ)/2. Alternatively, perhaps the optimal μ* is given by μ* = (σ^2 γ)/2. I think that's the most plausible answer, assuming r = 0. So, summarizing:1. For Investment A, r ≈ 16.09%.2. For Investment B, μ* = (σ^2 γ)/2.</think>