A popular random variable W for modeling the lifetime of a unit subject to failure, or...

 

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A popular random variable W for modeling the lifetime of a unit subject to failure, or the time between claims filing in actuarial science is the Weibull distribution (named after the Swedish mathematician Waloddi Weibull) with parameters a and 3, written W (a, 3) and with pdf given by _au^"- exp[- (#)"] awa- Ba f(w) = (a) Show that f(w) is indeed a valid pdf and find the edf F(w) of W. (b) Use the cdf of W to find the pth percentile of W, say up and find the median of W (c) What distribution correspond to a = 1? (d) Let X(w) be the instantaneous failure rate or hazard rate function [rate of failure in a small interval [w, w+dw) given survival to w] of a unit that follows a W(a, 3). The formula for X(w) is f(w) [1 - F(w)]* Find the hazard function of the Weibull distribution. (e) Show that X(w) is increasing for a > 1 and decreasing for a < 1. What are the implications of those results? X(w) = (f) Find the mean and variance of W(a, 3). These have to be expressed as a function of the Gamma function I(-). Recall that I(7 +1) = TI (7). (g) Take 3= 1 and plot in R or Matlab the graph of f(w) for a (0.3, 0.5, 1.2, 1.5,2} and comment on what you observe. Problem 3. The cost to Hedgehog Brewing of producing r Deluxe 2020 Hedgehog Special Edition Holiday Sampler Cases (D20HSEHSCs) is C(r) = 16000 + 500z-1.6x +0.0042 (measured in dollars), and market research (conducted by a crack team of hedgehog actuaries (these are literal hedgehogs trained in actuarial science at HMIT)) has shown that the maximum price Hedgehog can charge if they want to sell r D20HSEHSCs is p(z) = 1700-7x (measured in dollars per D20HSEHSC). (a) Determine the number of D20HSEHSCS Hedgehog should produce in order to maxi- mize profit. 1 (b) What is the price of a single D20HSEHSC at the production level you found in (a)? (c) How many D20HSEHSCs do you think Keenan can afford on his teacher's salary based on the price you gave in (b)? How many do you think he will actually buy? Problem 4. Hedgehog Brewing is preparing to brew its first ever collaboration beer. In order to shed its wholesome image as a family brewer and expand its customer base to include. fans of heavy metal, etc., Hedgehog will be working together with 3 Floyds Brewing (an esteemed brewer in Munster, Indiana that can reasonably be described as very metal) to produce an extremely limited run of a double black IPA (also known as a Cascadian dark ale) called Gazing Into the Abyss. The hedgehog actuaries (they are at it again!) predict that the marginal cost C'(x) to Hedgehog of producing a cases of Gazing Into the Abyss, in dollars, is given by 300 for 0x < 50, 250 for 50 x < 100, 220 for 100 150. C'(x)= (a) What are the units of C'(x)? (b) Sketch the graph of the marginal cost function C'. (c) Express the total cost to Hedgehog of producing 150 cases of Gazing Into the Abyss as an integral, and compute this integral. A popular random variable W for modeling the lifetime of a unit subject to failure, or the time between claims filing in actuarial science is the Weibull distribution (named after the Swedish mathematician Waloddi Weibull) with parameters a and 3, written W (a, 3) and with pdf given by _au^"- exp[- (#)"] awa- Ba f(w) = (a) Show that f(w) is indeed a valid pdf and find the edf F(w) of W. (b) Use the cdf of W to find the pth percentile of W, say up and find the median of W (c) What distribution correspond to a = 1? (d) Let X(w) be the instantaneous failure rate or hazard rate function [rate of failure in a small interval [w, w+dw) given survival to w] of a unit that follows a W(a, 3). The formula for X(w) is f(w) [1 - F(w)]* Find the hazard function of the Weibull distribution. (e) Show that X(w) is increasing for a > 1 and decreasing for a < 1. What are the implications of those results? X(w) = (f) Find the mean and variance of W(a, 3). These have to be expressed as a function of the Gamma function I(-). Recall that I(7 +1) = TI (7). (g) Take 3= 1 and plot in R or Matlab the graph of f(w) for a (0.3, 0.5, 1.2, 1.5,2} and comment on what you observe. Problem 3. The cost to Hedgehog Brewing of producing r Deluxe 2020 Hedgehog Special Edition Holiday Sampler Cases (D20HSEHSCs) is C(r) = 16000 + 500z-1.6x +0.0042 (measured in dollars), and market research (conducted by a crack team of hedgehog actuaries (these are literal hedgehogs trained in actuarial science at HMIT)) has shown that the maximum price Hedgehog can charge if they want to sell r D20HSEHSCs is p(z) = 1700-7x (measured in dollars per D20HSEHSC). (a) Determine the number of D20HSEHSCS Hedgehog should produce in order to maxi- mize profit. 1 (b) What is the price of a single D20HSEHSC at the production level you found in (a)? (c) How many D20HSEHSCs do you think Keenan can afford on his teacher's salary based on the price you gave in (b)? How many do you think he will actually buy? Problem 4. Hedgehog Brewing is preparing to brew its first ever collaboration beer. In order to shed its wholesome image as a family brewer and expand its customer base to include. fans of heavy metal, etc., Hedgehog will be working together with 3 Floyds Brewing (an esteemed brewer in Munster, Indiana that can reasonably be described as very metal) to produce an extremely limited run of a double black IPA (also known as a Cascadian dark ale) called Gazing Into the Abyss. The hedgehog actuaries (they are at it again!) predict that the marginal cost C'(x) to Hedgehog of producing a cases of Gazing Into the Abyss, in dollars, is given by 300 for 0x < 50, 250 for 50 x < 100, 220 for 100 150. C'(x)= (a) What are the units of C'(x)? (b) Sketch the graph of the marginal cost function C'. (c) Express the total cost to Hedgehog of producing 150 cases of Gazing Into the Abyss as an integral, and compute this integral.