A sum of $50,000 is invested at a rate R, selected from a

Problem 5E The pdf of X is f (x) = ? x??1, 0 < x < 1, 0 < ? < ?. Let Y = ?2? lnX. How is Y distributed?

Problem 1E Let X have the pdf f (x) = 4x3, 0 < x < 1. Find the pdf of Y = X2.

Let \(X\) have a logistic distribution with pdf \(f(x)=\frac{e^{-x}}{\left(1+e^{-x}\right)^{2}}, \quad-\infty<x<\infty\)/ . Show that \(Y=\frac{1}{1+e^{-X}}\) has a \(U(0,1)\) distribution. Equation Transcription: Text Transcription: X f(x)=e^-x/(1+e^-x)2^, - infinity <x< infinity Y=1/1+e^-X U(0,1)

Let \(X\) have a gamma distribution with \(\alpha=3\) and \(\theta=2\). Determine the pdf of \(Y=\sqrt{X}\). Equation Transcription: Text Transcription: X alpha=3 theta=2 Y=sqrt X

Problem 2E Let X have the pdf f (x) = xe?x2/2, 0 < x < ?. Find the pdf of Y = X2.

Problem 15E Let Y = X2. (a) Find the pdf of Y when the distribution of X is N(0, 1). (b) Find the pdf of Y when the pdf of X is f (x) = (3/2)x2, ? 1 < x < 1.

Problem 7E A sum of $50,000 is invested at a rate R, selected from a uniform distribution on the interval (0.03, 0.07). Once R is selected, the sum is compounded instantaneously for a year, so that X = 50000 eR dollars is the amount at the end of that year. (a) Find the cdf and pdf of X. (b) Verify that X = 50000 eR is defined correctly if the compounding is done instantaneously. Hint: Divide the year into n equal parts, calculate the value of the amount at the end of each part, and then take the limit as n??.

Problem 8E The lifetime (in years) of a manufactured product is Y = 5X0.7, where X has an exponential distribution with mean 1. Find the cdf and pdf of Y.

Problem 12E Let f (x) = 1/[?(1 + x2)], ?? < x < ?, be the pdf of the Cauchy random variable X. Show that E(X) does not exist.

If the distribution of \(X\) is \(N\left(\mu, \sigma^{2}\right)\), then \(M(t)=E\left(e^{t X}\right)=\exp \left(\mu t+\sigma^{2} t^{2} / 2\right)\). We then say that \(Y=e^{X}\) has a lognormal distribution because \(X=\ln Y\). (a) Show that the pdf of \(Y\) is \(g(y)=\frac{1}{y \sqrt{2 \pi \sigma^{2}}} \exp \left[-(\ln y-\mu)^{2} / 2 \sigma^{2}\right], \quad 0<y<\infty\) . (b) Using \(M(t)\), find (i) \(E(Y)=E\left(e^{X}\right)=M(1)\), (ii) \(E\left(Y^{2}\right)=E\left(e^{2 X}\right)=M(2)\), and (iii) \(\operatorname{Var}(Y)\). Equation Transcription: Text Transcription: X N( mu,2) M(t)=E(e^tX)=exp (mu t+sigma^2 t^2/2) Y=e^X X=ln?Y Y g(y)=1/y sqrt 2 pi sigma^2 exp[?-(ln?y-mu)^2/2 sigma^2], 0<y< infinity M(t) E(Y)=E(e^X)=M(1) E(Y^2)=E(e^2X)=M(2) Var?(Y)

Problem 10E Let X have the uniform distribution U(?1, 3). Find the pdf of Y = X2.

Problem 11E Let X have a Cauchy distribution. Find (a) P(X > 1). (b) P(X > 5). (c) P(X > 10).

Problem 14E Let X be N(0, 1). Find the pdf of Y = |X|, a distribution that is often called the half-normal. Hint: Here y ? Sy = {y : 0 < y < ?}. Consider the two transformations x1 = ?y,?? < x1 < 0, and x2 = y, 0 < y < ?.

Let X have the pdf f(x) = 4x3, 0 < x < 1. Find the pdf of Y = X2.

Let X have the pdf f(x) = xex2/2, 0 < x < . Find the pdf of Y = X2.

Let X have a gamma distribution with = 3 and = 2. Determine the pdf of Y = X.

The pdf of X is f(x) = 2x, 0 < x < 1. (a) Find the cdf of X. (b) Describe how an observation of X can be simulated. (c) Simulate 10 observations of X.

The pdf of X is \(f(x) = \theta x^{\theta-1}, 0 < x < 1, 0 <\theta < \infty\). Let \(Y = -2\theta \ln X\). How is Y distributed?

Let X have a logistic distribution with pdf \(f(x) = \frac {e^{-x}}{(1 + e?x)^2}\),\(\quad -\infty < x < \infty\). Show that \(Y = \frac {1}{1 + e^{-X}}\) has a U(0, 1) distribution.

A sum of $50,000 is invested at a rate R, selected from a uniform distribution on the interval (0.03, 0.07). Once R is selected, the sum is compounded instantaneously for a year, so that X = 50000 eR dollars is the amount at the end of that year. (a) Find the cdf and pdf of X. (b) Verify that X = 50000 eR is defined correctly if the compounding is done instantaneously. Hint: Divide the year into n equal parts, calculate the value of the amount at the end of each part, and then take the limit as n

The lifetime (in years) of a manufactured product is Y = 5X0.7, where X has an exponential distribution with mean 1. Find the cdf and pdf of Y.

Statisticians frequently use the extreme value distribution given by the cdf F(x) = 1 exp $ e(x 1)/2 % , < x < . A simple case is when 1 = 0 and 2 = 1, giving F(x) = 1 exp  ex , < x < . Let Y = eX or X = ln Y; then the support of Y is 0 < y < . (a) Show that the distribution of Y is exponential when 1 = 0 and 2 = 1. (b) Find the cdf and the pdf of Y when 1 = 0 and 2 > 0. (c) Let 1 = ln and 2 = 1/ in the cdf and pdf of Y. What is this distribution? (d) As suggested by its name, the extreme value distribution can be used to model the longest home run, the deepest mine, the greatest flood, and so on. Suppose the length X (in feet) of the maximum of someones home runs was modeled by an extreme value distribution with 1 = 550 and 2 = 25. What is the probability that X exceeds 500 feet?

Let X have the uniform distribution U(?1, 3). Find the pdf of \(Y = X^2\).

Let X have a Cauchy distribution. Find (a) P(X > 1). (b) P(X > 5). (c) P(X > 10).

Let f(x) = 1/[(1 + x2)], < x < , be the pdf of the Cauchy random variable X. Show that E(X) does not exist.

If the distribution of X is N(, 2), then M(t) = E(etX) = exp(t + 2t 2/2). We then say that Y = eX has a lognormal distribution because X = ln Y. (a) Show that the pdf of Y is g(y) = 1 y 22 exp[(ln y ) 2/22], 0 < y < . (b) Using M(t), find (i) E(Y) = E(eX) = M(1), (ii) E(Y2) = E(e2X) = M(2), and (iii) Var(Y).

Let X be N(0, 1). Find the pdf of Y = |X|, a distribution that is often called the half-normal. Hint: Here \(y~ \epsilon~ S_y = {y : 0 < y < \infty}\). Consider the two transformations \(x_1 = -y, -\infty < x_1 < 0\), and \(x_2 = y, 0 < y < \infty\).

Let Y = X2. (a) Find the pdf of Y when the distribution of X is N(0, 1). (b) Find the pdf of Y when the pdf of X is f(x) = (3/2)x2, 1 < x < 1.