🧠:<think>First, I need to determine the total savings for the university when switching to open-access textbooks. In the first year, each student spent an average of 500 on textbooks, and in the second year, this dropped to 150 per student. With 10,000 students, the total savings for the second year can be calculated by finding the difference in average costs and multiplying by the number of students.Next, I need to project the total savings for the next three years, considering that the university's enrollment increases by 5% annually. I'll use the compound interest formula to calculate the number of students each year. For each year, I'll compute the total savings by applying the same per-student savings and then sum these amounts to get the total projected savings over the three years.</think>

🧠:<think>Alright, so I'm trying to help this person who's considering laser eye surgery. They want to make an informed decision, so they've got two questions: one about probability and another about cost-benefit analysis. Let me break this down step by step.First, the probability analysis. They mentioned two types of surgeries: LASIK and PRK. The success rates are 0.95 for LASIK and 0.90 for PRK. The individual is considering both, but only the second if the first fails. They want the probability that at least one of the surgeries is successful.Hmm, okay. So, if I remember correctly, when dealing with probabilities of independent events, the probability that at least one occurs is equal to 1 minus the probability that neither occurs. That makes sense because it's easier to calculate the probability of both failing and subtracting that from 1.So, let's denote the success of LASIK as event A and the success of PRK as event B. Since they are independent, the probability that both fail is the product of their individual failure probabilities.The failure probability for LASIK is 1 - 0.95 = 0.05. For PRK, it's 1 - 0.90 = 0.10. So, the probability that both fail is 0.05 * 0.10 = 0.005.Therefore, the probability that at least one is successful is 1 - 0.005 = 0.995. So, that's 99.5%. That seems pretty high, which makes sense because both have high success rates.Now, moving on to the cost-benefit analysis. The costs are 2,000 for LASIK and 1,500 for PRK. The benefit of a successful surgery is 10,000, and it's received only once, meaning if either surgery is successful, the individual gets the benefit.So, we need to calculate the expected monetary value (EMV) of going through both surgeries. EMV is calculated by multiplying the probability of each outcome by its respective monetary value and then summing them all up.First, let's identify all possible outcomes and their probabilities:1. LASIK is successful. Probability = 0.95. In this case, they don't need PRK, so the cost is 2,000, and they receive the benefit of 10,000.2. LASIK fails, but PRK is successful. Probability = (1 - 0.95) * 0.90 = 0.05 * 0.90 = 0.045. Here, they pay for both surgeries, so the total cost is 2,000 + 1,500 = 3,500, and they receive the benefit of 10,000.3. Both LASIK and PRK fail. Probability = 0.05 * 0.10 = 0.005. They pay for both surgeries, total cost 3,500, and receive no benefit.So, now, let's compute the net monetary value for each outcome:1. LASIK successful: Net = Benefit - Cost = 10,000 - 2,000 = 8,000.2. LASIK fails, PRK successful: Net = 10,000 - 3,500 = 6,500.3. Both fail: Net = 0 - 3,500 = -3,500.Now, the EMV is the sum of each outcome's probability multiplied by its net value.So, EMV = (0.95 * 8,000) + (0.045 * 6,500) + (0.005 * -3,500).Let me calculate each term:First term: 0.95 * 8,000 = 7,600.Second term: 0.045 * 6,500. Let's see, 0.045 * 6,500. 0.04 * 6,500 is 260, and 0.005 * 6,500 is 32.5. So, total is 260 + 32.5 = 292.5.Third term: 0.005 * (-3,500) = -17.5.Now, adding them all together: 7,600 + 292.5 - 17.5.7,600 + 292.5 is 7,892.5. Then, subtracting 17.5 gives 7,875.So, the EMV is 7,875.Wait, let me double-check the calculations to make sure I didn't make a mistake.First term: 0.95 * 8,000. 0.95 * 8,000 is indeed 7,600.Second term: 0.045 * 6,500. Let me compute 6,500 * 0.045. 6,500 * 0.04 is 260, and 6,500 * 0.005 is 32.5, so total 292.5. Correct.Third term: 0.005 * (-3,500) is -17.5. Correct.Adding up: 7,600 + 292.5 = 7,892.5. Then, 7,892.5 - 17.5 = 7,875. Yes, that seems right.So, the expected monetary value is 7,875.Alternatively, another way to think about it is to compute the expected benefit and subtract the expected cost.Expected benefit: Probability of at least one success * 10,000. From the first part, that's 0.995 * 10,000 = 9,950.Expected cost: Probability of only LASIK * 2,000 + Probability of LASIK fail and PRK * 3,500 + Probability of both fail * 3,500.Wait, actually, no. The expected cost is the sum over all scenarios of the cost in each scenario multiplied by its probability.So, the cost scenarios:1. LASIK successful: Cost = 2,000, probability = 0.95.2. LASIK fails, PRK successful: Cost = 3,500, probability = 0.045.3. Both fail: Cost = 3,500, probability = 0.005.So, expected cost = (0.95 * 2,000) + (0.045 * 3,500) + (0.005 * 3,500).Calculating each term:0.95 * 2,000 = 1,900.0.045 * 3,500 = 157.5.0.005 * 3,500 = 17.5.Adding them up: 1,900 + 157.5 + 17.5 = 2,075.So, expected cost is 2,075.Expected benefit is 9,950.Therefore, EMV = Expected benefit - Expected cost = 9,950 - 2,075 = 7,875. Same result. So, that's consistent.Therefore, the EMV is 7,875.So, summarizing:1. Probability that at least one surgery is successful is 0.995 or 99.5%.2. The expected monetary value of undergoing both surgeries is 7,875.I think that's solid. I don't see any mistakes in the calculations. Both approaches gave the same result for the EMV, which is a good check.Final Answer1. The probability that at least one surgery will be successful is boxed{0.995}.2. The expected monetary value is boxed{7875} dollars.</think>

🧠:<think>Okay, so I have this problem where a language enthusiast is helping an artist communicate with a foreign audience. The artist's work uses a function f(x) to transform language input into visual art, and the audience's language is modeled by a probabilistic function p(y). The first part asks me to determine the value of T(y), which is defined as the integral from 0 to y of p(t) times f(t) dt. They give p(y) as a Gaussian distribution with mean μ and standard deviation σ, and f(x) as e^x. Alright, let me write down what I know. So, T(y) is the integral from 0 to y of p(t)f(t) dt. Given p(t) is (1/(sqrt(2πσ²))) e^(-(t - μ)²/(2σ²)) and f(t) is e^t. So, T(y) = ∫₀^y [1/(sqrt(2πσ²))] e^(-(t - μ)²/(2σ²)) * e^t dt.Hmm, that looks a bit complicated, but maybe I can simplify it. Let me see. The integrand is the product of a Gaussian and an exponential function. I wonder if there's a way to combine these exponents.Let me write the exponent part:-(t - μ)²/(2σ²) + t.Let me expand -(t - μ)²/(2σ²):= -(t² - 2μt + μ²)/(2σ²)= (-t² + 2μt - μ²)/(2σ²)= (-t²)/(2σ²) + (2μt)/(2σ²) - μ²/(2σ²)= (-t²)/(2σ²) + (μt)/σ² - μ²/(2σ²)So, adding t to this:Total exponent becomes:(-t²)/(2σ²) + (μt)/σ² - μ²/(2σ²) + tLet me combine like terms. The terms with t are (μ/σ² + 1) t. The quadratic term is (-1/(2σ²)) t², and the constant term is -μ²/(2σ²).So, the exponent is:(-1/(2σ²)) t² + (μ/σ² + 1) t - μ²/(2σ²)Hmm, that's still a quadratic in t. Maybe I can complete the square to make this integral more manageable.Let me denote the exponent as:A t² + B t + CWhere A = -1/(2σ²), B = (μ/σ² + 1), and C = -μ²/(2σ²).Completing the square for quadratic in t:A t² + B t + C = A(t² + (B/A) t) + C= A [ t² + (B/A) t + (B/(2A))² ] - A (B/(2A))² + C= A (t + B/(2A))² - B²/(4A) + CSo, plugging in A, B, C:= (-1/(2σ²)) [ t + ( (μ/σ² + 1) / (2*(-1/(2σ²))) ) ]² - ( (μ/σ² + 1) )² / (4*(-1/(2σ²))) + (-μ²/(2σ²))Let me compute each part step by step.First, the term inside the square:t + (B/(2A)) = t + [ (μ/σ² + 1) / (2*(-1/(2σ²))) ]Simplify denominator: 2*(-1/(2σ²)) = -1/σ²So, B/(2A) = (μ/σ² + 1) / (-1/σ²) = - (μ/σ² + 1) * σ² = -μ - σ²So, the square term is (t - μ - σ²)².Next, the coefficient A is -1/(2σ²), so:A (t + B/(2A))² = (-1/(2σ²)) (t - μ - σ²)²Now, the constant terms:- (B²)/(4A) + CCompute B²:(μ/σ² + 1)^2 = μ²/σ⁴ + 2μ/σ² + 1Divide by 4A:(μ²/σ⁴ + 2μ/σ² + 1) / (4*(-1/(2σ²))) = (μ²/σ⁴ + 2μ/σ² + 1) * (-2σ²/4) = (μ²/σ⁴ + 2μ/σ² + 1) * (-σ²/2)So, that's:- (μ²/σ⁴ * σ²/2) - (2μ/σ² * σ²/2) - (1 * σ²/2)Simplify each term:- μ²/(2σ²) - μ - σ²/2Adding the constant term C, which is -μ²/(2σ²):Total constant terms:[ - μ²/(2σ²) - μ - σ²/2 ] + [ - μ²/(2σ²) ] = - μ²/σ² - μ - σ²/2So, putting it all together, the exponent becomes:(-1/(2σ²)) (t - μ - σ²)² - μ²/σ² - μ - σ²/2Therefore, the integrand is:(1/(sqrt(2πσ²))) e^{ (-1/(2σ²)) (t - μ - σ²)² - μ²/σ² - μ - σ²/2 }We can write this as:(1/(sqrt(2πσ²))) e^{ - μ²/σ² - μ - σ²/2 } * e^{ (-1/(2σ²)) (t - μ - σ²)² }Notice that the term e^{ - μ²/σ² - μ - σ²/2 } is a constant with respect to t, so we can factor it out of the integral.Therefore, T(y) becomes:(1/(sqrt(2πσ²))) e^{ - μ²/σ² - μ - σ²/2 } ∫₀^y e^{ (-1/(2σ²)) (t - μ - σ²)² } dtNow, let me make a substitution to simplify the integral. Let u = t - μ - σ². Then, du = dt, and when t = 0, u = -μ - σ²; when t = y, u = y - μ - σ².So, the integral becomes:∫_{-μ - σ²}^{y - μ - σ²} e^{ (-1/(2σ²)) u² } duThat's the integral of a Gaussian function, which is related to the error function (erf). Specifically, the integral of e^{-a u²} du is (sqrt(π)/(2 sqrt(a))) erf(u sqrt(a)) + C.In our case, a = 1/(2σ²), so sqrt(a) = 1/(sqrt(2) σ). Therefore, the integral becomes:sqrt(2πσ²) * [ erf( (y - μ - σ²) / (sqrt(2) σ) ) - erf( (-μ - σ²) / (sqrt(2) σ) ) ] / 2Wait, let me verify that. The integral of e^{-a u²} du from u1 to u2 is (sqrt(π)/(2 sqrt(a))) [ erf(u2 sqrt(a)) - erf(u1 sqrt(a)) ]So, in our case, a = 1/(2σ²), so sqrt(a) = 1/(sqrt(2) σ). Therefore, the integral is:sqrt(π) / (2 * (1/(sqrt(2) σ))) ) [ erf( (y - μ - σ²) / (sqrt(2) σ) ) - erf( (-μ - σ²) / (sqrt(2) σ) ) ]Simplify the coefficient:sqrt(π) / (2 / (sqrt(2) σ)) ) = sqrt(π) * sqrt(2) σ / 2 = (sqrt(2π) σ)/2So, the integral becomes:(sqrt(2π) σ)/2 [ erf( (y - μ - σ²) / (sqrt(2) σ) ) - erf( (-μ - σ²) / (sqrt(2) σ) ) ]Now, putting this back into T(y):T(y) = (1/(sqrt(2πσ²))) e^{ - μ²/σ² - μ - σ²/2 } * (sqrt(2π) σ)/2 [ erf( (y - μ - σ²) / (sqrt(2) σ) ) - erf( (-μ - σ²) / (sqrt(2) σ) ) ]Simplify the constants:1/(sqrt(2πσ²)) = 1/(sqrt(2π) σ)Multiply by (sqrt(2π) σ)/2:(1/(sqrt(2π) σ)) * (sqrt(2π) σ)/2 = 1/2So, T(y) simplifies to:(1/2) e^{ - μ²/σ² - μ - σ²/2 } [ erf( (y - μ - σ²) / (sqrt(2) σ) ) - erf( (-μ - σ²) / (sqrt(2) σ) ) ]Hmm, that's a bit involved, but I think that's as simplified as it gets. Alternatively, we can factor out the constants in the exponent.Let me compute the exponent:- μ²/σ² - μ - σ²/2That's a constant term, so we can write it as e^{ - (μ² + μ σ² + σ²/2)/σ² }? Wait, no, let me compute it correctly.Wait, - μ²/σ² - μ - σ²/2 is just a constant, so we can write it as K = e^{ - μ²/σ² - μ - σ²/2 }Therefore, T(y) = K/2 [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - erf( (-μ - σ²)/(sqrt(2) σ) ) ]Alternatively, since erf is an odd function, erf(-x) = -erf(x), so the second term becomes -erf( (μ + σ²)/(sqrt(2) σ) )Therefore, T(y) = K/2 [ erf( (y - μ - σ²)/(sqrt(2) σ) ) + erf( (μ + σ²)/(sqrt(2) σ) ) ]But I'm not sure if that helps much. Maybe it's better to leave it as is.So, summarizing, T(y) is:(1/2) e^{ - μ²/σ² - μ - σ²/2 } [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - erf( (-μ - σ²)/(sqrt(2) σ) ) ]Alternatively, we can write it as:(1/2) e^{ - (μ² + μ σ² + σ²/2)/σ² } [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - erf( (-μ - σ²)/(sqrt(2) σ) ) ]Wait, let me check the exponent again:- μ²/σ² - μ - σ²/2It's not a single fraction, so maybe it's better to leave it as is.Alternatively, factor out -1/σ²:= - (μ² + μ σ² + σ⁴/2)/σ²Wait, no, because μ is not divided by σ². Hmm, maybe not helpful.Alternatively, perhaps we can write the exponent as:- (μ² + μ σ² + σ²/2)/σ² = - (μ² + μ σ² + (σ²)/2)/σ² = - (μ²/σ² + μ + 1/2 )Yes, that's correct:- μ²/σ² - μ - σ²/2 = - (μ²/σ² + μ + 1/2 )Wait, no:Wait, σ²/2 is just σ²/2, not (σ²)/2σ². So, actually, it's:- μ²/σ² - μ - σ²/2 = - (μ² + μ σ² + σ⁴/2)/σ²Wait, no, that's not correct. Let me compute:- μ²/σ² - μ - σ²/2 = - (μ² + μ σ² + σ⁴/2)/σ²Wait, let's see:Multiply numerator and denominator:= - [ μ² + μ σ² + (σ²)(σ²)/2 ] / σ²= - [ μ² + μ σ² + σ⁴/2 ] / σ²But that seems more complicated. Maybe it's better to just leave it as - μ²/σ² - μ - σ²/2.So, in conclusion, T(y) is:(1/2) e^{ - μ²/σ² - μ - σ²/2 } [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - erf( (-μ - σ²)/(sqrt(2) σ) ) ]I think that's as far as I can go without more specific information about μ and σ.Wait, but maybe I made a mistake earlier. Let me double-check the substitution.We had:T(y) = ∫₀^y p(t) f(t) dt = ∫₀^y (1/(sqrt(2πσ²))) e^{ -(t - μ)^2/(2σ²) } e^t dtThen, combining exponents:= (1/(sqrt(2πσ²))) ∫₀^y e^{ - (t - μ)^2/(2σ²) + t } dtWhich I expanded to:= (1/(sqrt(2πσ²))) ∫₀^y e^{ (-t² + 2μt - μ²)/(2σ²) + t } dt= (1/(sqrt(2πσ²))) ∫₀^y e^{ (-t² + 2μt - μ² + 2σ² t)/(2σ²) } dtWait, hold on, I think I missed a step here. When combining exponents, it's:- (t - μ)^2/(2σ²) + t = - (t² - 2μt + μ²)/(2σ²) + t= - t²/(2σ²) + μt/σ² - μ²/(2σ²) + tSo, combining the linear terms in t:= (-t²)/(2σ²) + (μ/σ² + 1) t - μ²/(2σ²)Yes, that's correct. So, then completing the square as I did before.So, I think my earlier steps are correct.Therefore, the final expression for T(y) is:(1/2) e^{ - μ²/σ² - μ - σ²/2 } [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - erf( (-μ - σ²)/(sqrt(2) σ) ) ]Alternatively, we can write this as:(1/2) e^{ - (μ² + μ σ² + σ²/2)/σ² } [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - erf( (-μ - σ²)/(sqrt(2) σ) ) ]But I think the first form is clearer.So, that's the answer for part 1.Now, moving on to part 2. We need to find the interval [a, b] such that the average of T(y) over [a, b] is 100. That is:(1/(b - a)) ∫ₐ^b T(y) dy = 100Given that T(y) is the function we found in part 1, which is:T(y) = (1/2) e^{ - μ²/σ² - μ - σ²/2 } [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - erf( (-μ - σ²)/(sqrt(2) σ) ) ]So, we need to compute the integral of T(y) from a to b, divide by (b - a), and set it equal to 100. Then solve for a and b.But this seems quite complicated because T(y) itself is an integral involving the error function. Integrating T(y) again would lead to even more complex expressions.Alternatively, perhaps we can find an expression for the integral of T(y) over [a, b] in terms of known functions.Let me denote:Let’s denote C = (1/2) e^{ - μ²/σ² - μ - σ²/2 } and D = erf( (-μ - σ²)/(sqrt(2) σ) )So, T(y) = C [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - D ]Therefore, ∫ₐ^b T(y) dy = C ∫ₐ^b [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - D ] dy= C [ ∫ₐ^b erf( (y - μ - σ²)/(sqrt(2) σ) ) dy - D (b - a) ]So, we need to compute ∫ₐ^b erf( (y - μ - σ²)/(sqrt(2) σ) ) dyLet me make a substitution to simplify this integral. Let u = (y - μ - σ²)/(sqrt(2) σ). Then, y = sqrt(2) σ u + μ + σ², and dy = sqrt(2) σ du.When y = a, u = (a - μ - σ²)/(sqrt(2) σ) = let's call this u_aWhen y = b, u = (b - μ - σ²)/(sqrt(2) σ) = u_bSo, the integral becomes:∫_{u_a}^{u_b} erf(u) * sqrt(2) σ duWe know that the integral of erf(u) du is (2/sqrt(π)) u e^{u²} - (2/sqrt(π)) ∫ e^{u²} du, but wait, actually, the integral of erf(u) is (2/sqrt(π)) e^{u²} + C. Wait, no, let me recall:The integral of erf(u) du is (2/sqrt(π)) u e^{u²} - (2/sqrt(π)) ∫ e^{u²} du, but ∫ e^{u²} du doesn't have an elementary form, it's related to the imaginary error function.Wait, actually, I think the integral of erf(u) du is (2/sqrt(π)) (u e^{u²} - ∫ e^{u²} du ). But ∫ e^{u²} du is not elementary, so perhaps we can express it in terms of the error function.Wait, let me check:Let me recall that ∫ erf(u) du = (2/sqrt(π)) (u e^{u²} - ∫ e^{u²} du )But ∫ e^{u²} du is (sqrt(π)/2) erf(i u) + C, but that's involving imaginary error functions, which complicates things.Alternatively, perhaps we can express the integral in terms of the original variables.Wait, maybe another approach. Let me consider integrating T(y) over [a, b].Given that T(y) is the integral from 0 to y of p(t) f(t) dt, then integrating T(y) from a to b is the same as integrating p(t) f(t) over the region where t ≤ y, and y ranges from a to b.This is a double integral, which can be expressed as:∫ₐ^b T(y) dy = ∫ₐ^b [ ∫₀^y p(t) f(t) dt ] dyWe can switch the order of integration. The region of integration is 0 ≤ t ≤ y ≤ b, and a ≤ y ≤ b. So, switching the order, t goes from 0 to b, and for each t, y goes from max(t, a) to b.Therefore:∫ₐ^b T(y) dy = ∫₀^b p(t) f(t) [ ∫_{max(t, a)}^b dy ] dt= ∫₀^b p(t) f(t) (b - max(t, a)) dt= ∫₀^a p(t) f(t) (b - a) dt + ∫_a^b p(t) f(t) (b - t) dtSo, that's a different expression for the integral of T(y) over [a, b]. Maybe this is easier to compute.Given that p(t) f(t) is (1/(sqrt(2πσ²))) e^{ -(t - μ)^2/(2σ²) } e^t = (1/(sqrt(2πσ²))) e^{ - (t - μ)^2/(2σ²) + t }Which is the same integrand as before. So, perhaps we can compute these integrals.Let me denote:I1 = ∫₀^a p(t) f(t) dtI2 = ∫_a^b p(t) f(t) (b - t) dtThen, ∫ₐ^b T(y) dy = (b - a) I1 + I2But I1 is T(a), since T(a) = ∫₀^a p(t) f(t) dtSimilarly, I2 is ∫_a^b (b - t) p(t) f(t) dtSo, putting it all together:∫ₐ^b T(y) dy = (b - a) T(a) + ∫_a^b (b - t) p(t) f(t) dtHmm, but I'm not sure if that helps directly. Alternatively, perhaps we can compute I2 by integrating by parts.Let me try integrating I2 by parts. Let u = b - t, dv = p(t) f(t) dtThen, du = -dt, and v = T(t) - T(a) since v = ∫ p(t) f(t) dt from a to t.Wait, actually, v = ∫ p(t) f(t) dt from a to t, which is T(t) - T(a)So, integrating by parts:I2 = u v | from a to b - ∫ₐ^b v du= [ (b - t)(T(t) - T(a)) ] from a to b - ∫ₐ^b (T(t) - T(a)) (-1) dtEvaluate the boundary term:At t = b: (b - b)(T(b) - T(a)) = 0At t = a: (b - a)(T(a) - T(a)) = 0So, the boundary term is 0.Therefore, I2 = ∫ₐ^b (T(t) - T(a)) dtSo, ∫ₐ^b T(y) dy = (b - a) T(a) + ∫ₐ^b (T(t) - T(a)) dt= (b - a) T(a) + ∫ₐ^b T(t) dt - (b - a) T(a)= ∫ₐ^b T(t) dtWait, that's just restating the original integral. Hmm, that didn't help.Alternatively, perhaps we can express I2 in terms of T(t):I2 = ∫_a^b (b - t) p(t) f(t) dtLet me make a substitution: let s = b - t, then when t = a, s = b - a, and when t = b, s = 0. So, dt = -ds.Thus, I2 = ∫_{b - a}^0 s p(b - s) f(b - s) (-ds) = ∫₀^{b - a} s p(b - s) f(b - s) dsBut I'm not sure if that helps.Alternatively, perhaps we can consider that p(t) f(t) is the same integrand as before, so maybe we can express I2 in terms of T(t) and some other function.Wait, perhaps another approach. Let me recall that T(y) is the integral from 0 to y of p(t) f(t) dt, so T'(y) = p(y) f(y). Therefore, integrating T(y) over [a, b] is ∫ₐ^b T(y) dy, which is the same as ∫ₐ^b ∫₀^y p(t) f(t) dt dy.As I did before, switching the order of integration, we get:∫₀^b p(t) f(t) ∫_{max(t, a)}^b dy dt = ∫₀^a p(t) f(t) (b - a) dt + ∫_a^b p(t) f(t) (b - t) dtSo, that's the same as before.Therefore, ∫ₐ^b T(y) dy = (b - a) ∫₀^a p(t) f(t) dt + ∫_a^b (b - t) p(t) f(t) dt= (b - a) T(a) + ∫_a^b (b - t) p(t) f(t) dtBut I still don't see an easy way to compute this without knowing specific values for a and b.Given that the average is 100, we have:(1/(b - a)) [ (b - a) T(a) + ∫_a^b (b - t) p(t) f(t) dt ] = 100Simplify:T(a) + (1/(b - a)) ∫_a^b (b - t) p(t) f(t) dt = 100Hmm, but I'm not sure how to proceed from here. Maybe we can consider that p(t) f(t) is a known function, and perhaps we can express the integral in terms of T(t) or other functions.Wait, let me recall that p(t) f(t) is the derivative of T(t). So, T'(t) = p(t) f(t). Therefore, ∫ (b - t) p(t) f(t) dt = ∫ (b - t) T'(t) dtIntegrate by parts:Let u = b - t, dv = T'(t) dtThen, du = -dt, v = T(t)So, ∫ (b - t) T'(t) dt = (b - t) T(t) | from a to b - ∫ T(t) (-1) dt= [ (b - b) T(b) - (b - a) T(a) ] + ∫ₐ^b T(t) dt= 0 - (b - a) T(a) + ∫ₐ^b T(t) dtTherefore, ∫ₐ^b (b - t) p(t) f(t) dt = - (b - a) T(a) + ∫ₐ^b T(t) dtBut we already have ∫ₐ^b T(t) dt = (b - a) T(a) + ∫ₐ^b (b - t) p(t) f(t) dtWait, substituting back:∫ₐ^b T(t) dt = (b - a) T(a) + [ - (b - a) T(a) + ∫ₐ^b T(t) dt ]Simplify:∫ₐ^b T(t) dt = (b - a) T(a) - (b - a) T(a) + ∫ₐ^b T(t) dtWhich simplifies to:∫ₐ^b T(t) dt = ∫ₐ^b T(t) dtWhich is just an identity, so it doesn't help.Hmm, perhaps another approach. Let me consider that T(y) is the integral of p(t) f(t) from 0 to y, and p(t) f(t) is a known function. So, perhaps I can express the integral of T(y) over [a, b] in terms of T(y) and its integral.Alternatively, perhaps I can consider that the average of T(y) over [a, b] is 100, which is a constant. So, maybe T(y) is approximately constant over [a, b], but that's only true if T(y) is flat, which it's not unless p(t) f(t) is zero, which it's not.Alternatively, perhaps we can assume that [a, b] is symmetric around some point, but without more information, it's hard to say.Alternatively, maybe we can consider that the integral of T(y) over [a, b] is 100*(b - a). So, we have:∫ₐ^b T(y) dy = 100 (b - a)But T(y) is given by the expression we found earlier. So, perhaps we can set up the equation:(1/(b - a)) ∫ₐ^b T(y) dy = 100Which is:∫ₐ^b T(y) dy = 100 (b - a)But T(y) is a function involving the error function, so integrating it again would be difficult. Maybe we can consider specific values or make approximations.Alternatively, perhaps we can consider that T(y) is a cumulative distribution function scaled by some constants, so the integral of T(y) over [a, b] might relate to some probability.Wait, T(y) is the integral from 0 to y of p(t) f(t) dt. So, T(y) is the expected value of f(t) up to y, scaled by some constants.But I'm not sure. Alternatively, maybe we can consider that T(y) is related to the convolution of p(t) and f(t), but I'm not sure.Alternatively, perhaps we can make a substitution to simplify T(y). Let me recall that T(y) is:T(y) = (1/2) e^{ - μ²/σ² - μ - σ²/2 } [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - erf( (-μ - σ²)/(sqrt(2) σ) ) ]Let me denote:Let’s define z = (y - μ - σ²)/(sqrt(2) σ)Then, y = sqrt(2) σ z + μ + σ²So, T(y) can be written as:T(y) = C [ erf(z) - erf(z0) ]Where z0 = (-μ - σ²)/(sqrt(2) σ)And C = (1/2) e^{ - μ²/σ² - μ - σ²/2 }So, T(y) = C [ erf(z) - erf(z0) ]Therefore, the integral of T(y) over [a, b] is:∫ₐ^b C [ erf(z) - erf(z0) ] dy= C ∫ₐ^b [ erf(z) - erf(z0) ] dyBut z is a function of y, so we can write:= C ∫_{z_a}^{z_b} [ erf(z) - erf(z0) ] * (sqrt(2) σ dz )Where z_a = (a - μ - σ²)/(sqrt(2) σ), z_b = (b - μ - σ²)/(sqrt(2) σ)So, the integral becomes:C sqrt(2) σ ∫_{z_a}^{z_b} [ erf(z) - erf(z0) ] dzNow, integrating [ erf(z) - erf(z0) ] dz:= ∫ erf(z) dz - erf(z0) ∫ dz= (2/sqrt(π)) z e^{z²} - (2/sqrt(π)) ∫ e^{z²} dz - erf(z0) z + CBut again, ∫ e^{z²} dz is not elementary, so we might need to express it in terms of the imaginary error function or leave it as is.Alternatively, perhaps we can express the integral in terms of T(y) and other known functions.But this seems too involved, and I'm not sure if there's a closed-form solution. Therefore, perhaps the best approach is to state that the interval [a, b] must satisfy the equation:∫ₐ^b T(y) dy = 100 (b - a)Where T(y) is given by the expression we found earlier. Therefore, the interval [a, b] must be such that this equation holds. The existence of such an interval depends on the behavior of T(y). Since T(y) is an integral of a positive function (p(t) f(t) is positive because p(t) is a probability density and f(t) = e^t is positive), T(y) is an increasing function. Therefore, T(y) is monotonically increasing. Therefore, the integral of T(y) over [a, b] is also increasing in both a and b. Therefore, for a given average value of 100, there may be a unique interval [a, b] that satisfies the condition, provided that T(y) achieves the necessary values.Alternatively, perhaps we can consider that since T(y) is increasing, the average over [a, b] is between T(a) and T(b). Therefore, if 100 is between T(a) and T(b), then such an interval exists. But since T(y) is increasing, T(a) ≤ 100 ≤ T(b) would be required. But without knowing the specific values of μ and σ, it's hard to say more.Alternatively, perhaps we can consider that the average of T(y) over [a, b] is equal to the integral of T(y) over [a, b] divided by (b - a). Since T(y) is increasing, the average is between T(a) and T(b). Therefore, if 100 is between T(a) and T(b), then such an interval exists. But since T(y) can be made arbitrarily large by choosing large y, and can be made small by choosing small y, there must exist some interval [a, b] where the average is 100, provided that T(y) is continuous and covers the value 100 in its range.But since T(y) is continuous and increasing, and as y approaches infinity, T(y) approaches some limit, and as y approaches negative infinity, T(y) approaches another limit. Therefore, depending on the specific form of T(y), the average might or might not reach 100.Wait, let me think about the limits. As y approaches infinity, T(y) approaches the integral from 0 to infinity of p(t) f(t) dt. Since p(t) is a Gaussian, and f(t) = e^t, the integral might converge or diverge depending on the parameters.Wait, p(t) is a Gaussian centered at μ with variance σ², and f(t) = e^t. So, the integrand is e^{ - (t - μ)^2/(2σ²) + t }.Let me analyze the behavior as t approaches infinity. The exponent is - (t² - 2μt + μ²)/(2σ²) + t = - t²/(2σ²) + (μ/σ² + 1) t - μ²/(2σ²)As t approaches infinity, the dominant term is - t²/(2σ²), which goes to negative infinity. Therefore, the integrand decays exponentially, so the integral converges.Similarly, as t approaches negative infinity, the exponent is dominated by - t²/(2σ²), which also goes to negative infinity, so the integrand decays as well. Therefore, T(y) approaches a finite limit as y approaches infinity.Therefore, T(y) is bounded above by some finite value, say M = ∫₀^∞ p(t) f(t) dt.Similarly, as y approaches negative infinity, T(y) approaches ∫₀^{-∞} p(t) f(t) dt, which is negative infinity? Wait, no, because p(t) is a Gaussian, which is positive everywhere, and f(t) = e^t is positive everywhere. Therefore, T(y) is the integral from 0 to y of a positive function. So, if y is negative, the integral from 0 to y is negative, but since p(t) f(t) is positive, T(y) is negative for y < 0.Wait, but in our case, y is a random variable with a Gaussian distribution, so y can take any real value. But in the transformation function T(y), y is the upper limit of integration, so y can be any real number.But in the average intensity problem, we are looking for an interval [a, b] where the average of T(y) is 100. Since T(y) is bounded above by M and can go to negative infinity as y approaches negative infinity, but actually, since p(t) f(t) is positive, T(y) is increasing and approaches M as y approaches infinity, and approaches some finite value as y approaches negative infinity? Wait, no, because as y approaches negative infinity, the integral from 0 to y is negative, but p(t) f(t) is positive, so T(y) approaches negative infinity.Wait, no, because p(t) is a Gaussian, which is symmetric, but f(t) = e^t is not symmetric. So, as y approaches negative infinity, the integral from 0 to y of p(t) f(t) dt would be negative, but does it approach negative infinity?Wait, let me consider the integral ∫₀^y p(t) f(t) dt as y approaches negative infinity. Since p(t) f(t) is positive for all t, the integral from 0 to y (where y < 0) is negative, and as y approaches negative infinity, the integral approaches negative infinity because the area under p(t) f(t) from 0 to y becomes more and more negative.Wait, no, actually, p(t) f(t) is positive for all t, so the integral from 0 to y is negative when y < 0, but does it approach negative infinity? Let me check.As y approaches negative infinity, the integral ∫₀^y p(t) f(t) dt = - ∫_y^0 p(t) f(t) dt. Since p(t) f(t) is positive, ∫_y^0 p(t) f(t) dt approaches ∫_{-infty}^0 p(t) f(t) dt, which is a finite value because p(t) f(t) decays exponentially as t approaches negative infinity.Wait, let me compute ∫_{-infty}^0 p(t) f(t) dt.p(t) f(t) = (1/(sqrt(2πσ²))) e^{ - (t - μ)^2/(2σ²) + t }Let me make a substitution: let u = t - μ, so t = u + μ, dt = du.Then, the exponent becomes:- u²/(2σ²) + u + μSo, p(t) f(t) = (1/(sqrt(2πσ²))) e^{ - u²/(2σ²) + u + μ }= (1/(sqrt(2πσ²))) e^{μ} e^{ - u²/(2σ²) + u }Now, let me complete the square in the exponent:- u²/(2σ²) + u = - (u² - 2σ² u)/(2σ²) = - (u² - 2σ² u + σ⁴ - σ⁴)/(2σ²) = - ( (u - σ²)^2 - σ⁴ )/(2σ² ) = - (u - σ²)^2/(2σ²) + σ²/2Therefore, p(t) f(t) = (1/(sqrt(2πσ²))) e^{μ} e^{ - (u - σ²)^2/(2σ²) + σ²/2 }= (1/(sqrt(2πσ²))) e^{μ + σ²/2} e^{ - (u - σ²)^2/(2σ²) }So, ∫_{-infty}^0 p(t) f(t) dt = ∫_{-infty}^{-μ} (1/(sqrt(2πσ²))) e^{μ + σ²/2} e^{ - (u - σ²)^2/(2σ²) } duWait, no, when t approaches 0, u = t - μ approaches -μ. So, when t = 0, u = -μ. Therefore, the integral becomes:∫_{-infty}^{-μ} (1/(sqrt(2πσ²))) e^{μ + σ²/2} e^{ - (u - σ²)^2/(2σ²) } duLet me make another substitution: let v = u - σ². Then, u = v + σ², du = dv.When u = -infty, v = -infty; when u = -μ, v = -μ - σ².So, the integral becomes:∫_{-infty}^{-μ - σ²} (1/(sqrt(2πσ²))) e^{μ + σ²/2} e^{ - v²/(2σ²) } dv= e^{μ + σ²/2} ∫_{-infty}^{-μ - σ²} (1/(sqrt(2πσ²))) e^{ - v²/(2σ²) } dv= e^{μ + σ²/2} Φ( (-μ - σ²)/σ )Where Φ is the cumulative distribution function of the standard normal distribution.But Φ(z) = (1/2) [1 + erf(z / sqrt(2))]Therefore, ∫_{-infty}^0 p(t) f(t) dt = e^{μ + σ²/2} Φ( (-μ - σ²)/σ )Which is a finite value. Therefore, T(y) approaches this finite value as y approaches negative infinity.Similarly, as y approaches infinity, T(y) approaches ∫₀^∞ p(t) f(t) dt, which is:∫₀^∞ p(t) f(t) dt = e^{μ + σ²/2} [1 - Φ( (-μ - σ²)/σ ) ]Therefore, T(y) is bounded between two finite values: T(-infty) = e^{μ + σ²/2} Φ( (-μ - σ²)/σ ) and T(infty) = e^{μ + σ²/2} [1 - Φ( (-μ - σ²)/σ ) ]Therefore, the range of T(y) is [ T(-infty), T(infty) ]Therefore, for the average of T(y) over [a, b] to be 100, 100 must lie within the range of T(y). That is, T(-infty) ≤ 100 ≤ T(infty)So, the condition for the existence of such an interval [a, b] is that 100 is between T(-infty) and T(infty). If 100 is outside this range, then no such interval exists.Given that T(y) is continuous and strictly increasing (since p(t) f(t) is positive), for any value between T(-infty) and T(infty), there exists some y such that T(y) equals that value. Therefore, by the Intermediate Value Theorem, for any average value between T(-infty) and T(infty), there exists an interval [a, b] such that the average of T(y) over [a, b] is 100.But wait, the average is not just a single value of T(y), but the integral over [a, b] divided by (b - a). So, even though T(y) is bounded, the average could potentially be outside the range of T(y). For example, if T(y) is always less than 100, then the average can't be 100. Similarly, if T(y) is always greater than 100, the average can't be 100.Wait, but since T(y) is increasing, the average over [a, b] is between T(a) and T(b). Therefore, if 100 is between T(a) and T(b), then such an interval exists. But since T(y) is bounded, if 100 is outside the range of T(y), then no such interval exists.Therefore, the condition for the existence of such an interval [a, b] is that 100 must lie within the range of T(y), i.e., T(-infty) ≤ 100 ≤ T(infty)Given that T(-infty) = e^{μ + σ²/2} Φ( (-μ - σ²)/σ )And T(infty) = e^{μ + σ²/2} [1 - Φ( (-μ - σ²)/σ ) ]Therefore, 100 must satisfy:e^{μ + σ²/2} Φ( (-μ - σ²)/σ ) ≤ 100 ≤ e^{μ + σ²/2} [1 - Φ( (-μ - σ²)/σ ) ]So, that's the condition for the existence of such an interval [a, b].Therefore, to summarize:1. T(y) is given by the expression involving the error function as derived earlier.2. The interval [a, b] exists if and only if 100 is within the range of T(y), i.e., 100 is between T(-infty) and T(infty). The exact values of a and b would depend on solving the integral equation, which may not have a closed-form solution and would likely require numerical methods.So, the final answers are:1. T(y) = (1/2) e^{ - μ²/σ² - μ - σ²/2 } [ erf( (y - μ - σ²)/(sqrt(2) σ) ) - erf( (-μ - σ²)/(sqrt(2) σ) ) ]2. The interval [a, b] exists if and only if 100 is between T(-infty) and T(infty), which are e^{μ + σ²/2} Φ( (-μ - σ²)/σ ) and e^{μ + σ²/2} [1 - Φ( (-μ - σ²)/σ ) ] respectively.</think>

🧠:<think>Alright, so I have this problem about a property developer who wants to list properties on two different platforms, Platform A and Platform B. The developer has 10 properties with different expected sale prices, and I need to figure out which platform is more cost-effective. Let me break this down step by step.First, let me understand the costs involved for each platform. Platform A charges a fixed monthly fee of 500 and an additional 1.5% commission on each property's sale price. Platform B, on the other hand, doesn't have a monthly fee but takes a higher commission of 3.5% on each sale. The developer is planning to list all 10 properties on both platforms, so I need to calculate the total cost for each platform and then compare them.Starting with Platform A. The total cost will have two components: the fixed monthly fee and the commission on each sale. Since the developer is listing 10 properties, I need to calculate the commission for each property and then sum them up, adding the fixed fee at the end.Let me write down the formula for the total cost on Platform A:Total Cost A = Fixed Fee + (Commission Rate A * Sum of All Sale Prices)Given that the fixed fee is 500, and the commission rate is 1.5%, which is 0.015 in decimal. So, I need to calculate the sum of all the sale prices first.Looking at the sale prices:P1 = 250,000P2 = 300,000P3 = 350,000P4 = 400,000P5 = 450,000P6 = 500,000P7 = 550,000P8 = 600,000P9 = 650,000P10 = 700,000I can add these up one by one. Let me do that:Start with P1: 250,000Add P2: 250,000 + 300,000 = 550,000Add P3: 550,000 + 350,000 = 900,000Add P4: 900,000 + 400,000 = 1,300,000Add P5: 1,300,000 + 450,000 = 1,750,000Add P6: 1,750,000 + 500,000 = 2,250,000Add P7: 2,250,000 + 550,000 = 2,800,000Add P8: 2,800,000 + 600,000 = 3,400,000Add P9: 3,400,000 + 650,000 = 4,050,000Add P10: 4,050,000 + 700,000 = 4,750,000So, the total sum of all sale prices is 4,750,000.Now, calculating the commission for Platform A:Commission A = 0.015 * 4,750,000Let me compute that:0.015 * 4,750,000 = ?Well, 1% of 4,750,000 is 47,500. So, 0.015 is 1.5%, which is 47,500 * 1.5.Calculating 47,500 * 1.5:47,500 * 1 = 47,50047,500 * 0.5 = 23,750Adding them together: 47,500 + 23,750 = 71,250So, the commission for Platform A is 71,250.Adding the fixed fee of 500:Total Cost A = 71,250 + 500 = 71,750Alright, so Platform A's total cost is 71,750.Now, moving on to Platform B. Platform B doesn't have a fixed fee, so the total cost will only be the commission, which is 3.5% on each sale. So, the formula for Platform B is:Total Cost B = Commission Rate B * Sum of All Sale PricesCommission Rate B is 3.5%, which is 0.035 in decimal.We already calculated the sum of all sale prices as 4,750,000.So, Commission B = 0.035 * 4,750,000Let me compute that:0.035 * 4,750,000First, 0.01 * 4,750,000 = 47,500So, 0.03 * 4,750,000 = 3 * 47,500 = 142,5000.005 * 4,750,000 = 23,750Adding 142,500 + 23,750 = 166,250So, the commission for Platform B is 166,250.Since there's no fixed fee, the total cost for Platform B is 166,250.Now, comparing the two total costs:Platform A: 71,750Platform B: 166,250Clearly, Platform A is cheaper. The difference is 166,250 - 71,750 = 94,500.So, Platform A is more cost-effective by 94,500.Wait, let me double-check my calculations to make sure I didn't make any errors.First, the sum of sale prices:250k + 300k = 550k550k + 350k = 900k900k + 400k = 1,300k1,300k + 450k = 1,750k1,750k + 500k = 2,250k2,250k + 550k = 2,800k2,800k + 600k = 3,400k3,400k + 650k = 4,050k4,050k + 700k = 4,750kYes, that's correct.Commission for A: 1.5% of 4,750k.1.5% is 0.015.0.015 * 4,750,000 = 71,250. Correct.Plus fixed fee: 71,250 + 500 = 71,750. Correct.Commission for B: 3.5% of 4,750k.3.5% is 0.035.0.035 * 4,750,000 = 166,250. Correct.So, yes, Platform A is cheaper.Alternatively, another way to think about it is to compare the commission rates. Platform A has a lower commission rate but a fixed fee, while Platform B has a higher commission rate but no fixed fee. Since the developer is selling 10 properties with a total sale price of 4.75 million, the fixed fee of 500 is negligible compared to the potential savings from the lower commission rate.To see if the fixed fee is worth it, let's calculate the break-even point where both platforms cost the same.Let me denote:Total Cost A = 500 + 0.015 * Total SalesTotal Cost B = 0.035 * Total SalesSet them equal:500 + 0.015 * S = 0.035 * SSubtract 0.015 * S from both sides:500 = 0.02 * SSo, S = 500 / 0.02 = 25,000Wait, that can't be right. Wait, 500 / 0.02 is 25,000? Wait, 0.02 * 25,000 = 500, yes.But in our case, the total sales are 4,750,000, which is way higher than 25,000. So, for total sales above 25,000, Platform A is cheaper because the fixed fee is offset by the lower commission rate.But in reality, the break-even point is when the fixed fee equals the difference in commission rates times the sales.Wait, let me think again.The difference in commission rates is 0.035 - 0.015 = 0.02.So, the fixed fee is 500, so the break-even sales would be 500 / 0.02 = 25,000.So, if the total sales are more than 25,000, Platform A is cheaper. Since our total sales are 4,750,000, which is way more than 25,000, Platform A is definitely cheaper.So, that confirms my earlier calculation.Therefore, Platform A is more cost-effective.Final AnswerThe total cost for Platform A is boxed{71750} dollars, making Platform A the more cost-effective option.</think>