Let X have a beta distribution with parameters and . (See Example 5.2-3.) (a) Show that

In Example 5.2-6, verify that the given transformation maps \(\left\{\left(x_{1}, x_{2}\right): 0<x_{1}<1,0<x_{2}<1\right\}\) onto \(\left\{\left(z_{1}, z_{2}\right):-\infty<z_{1}<\infty,-\infty<z_{2}<\infty\right\}\), except for a set of points that has probability 0 . HINT: What is the image of vertical line segments? What is the image of horizontal line segments? Equation Transcription: Text Transcription: {(x_1,x_2):0<x_1<1,0<x_2<1} {(z_1,z_2):-infinity <z_1<infinity ,-infinity <z_2<infinity}

Problem 1E Let X1, X2 denote two independent random variables, each with a ?2(2) distribution. Find the joint pdf of Y1 = X1 and Y2 = X2 + X1. Note that the support of Y1, Y2 is 0 < y1 < y2 < ?. Also, find the marginal pdf of each of Y1 and Y2. Are Y1 and Y2 independent?

Problem 2E Let X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Let Y1 = (X1/r1)/(X2/r2) and Y2 = X2. (a) Find the joint pdf of Y1 and Y2. (b) Determine the marginal pdf of Y1 and show that Y1 has an F distribution. (This is another, but equivalent, way of finding the pdf of F.)

Find the mean and the variance of an F random variable with and degrees of freedom by first finding E(U), E(1/V), E( ), and E(1/ ).

Problem 4E Let the distribution of W be F(9, 24). Find the following: (a) F 0.05(9, 24). (b) F 0.95(9, 24). (c) P(0.277 ? W ? 2.70).

Problem 5E Let the distribution of W be F(8, 4). Find the following: (a) F 0.01(8, 4). (b) F 0.99(8, 4). (c) P(0.198 ? W ? 8.98).

Let \(X_{1}\) and \(X_{2}\) be independent chi-square random variables with \(r_{1}\) and \(r_{2}\) degrees of freedom, respectively. Show that (a) \(U=X_{1} /\left(X_{1}+X_{2}\right)\) has a beta distribution with \(\alpha=r_{1} / 2\) and \(\beta=r_{2} / 2\). (b) \(V=X_{2} /\left(X_{1}+X_{2}\right)\) has a beta distribution with \(\alpha=r_{2} / 2\) and \(\beta=r_{1} / 2\). Equation Transcription: Text Transcription: X_1 X_2 r_1 r_2 U=X_1/X_1+X_2 alpha=r_1/2 beta=r_2/2 V=X_2/(X_1+X_2) alpha=r_2/2 beta=r_1/2

Problem 9E Determine the constant c such that f (x) = cx3(1 ? x)6, 0 < x < 1, is a pdf.

Let \(X\) have a beta distribution with parameters \(\alpha\) and \(\beta\). (See Example 5.2-3.) (a) Show that the mean and variance of \(X\) are, respectively, \(\mu=\frac{\alpha}{\alpha+\beta} \quad \text { and } \quad \sigma^{2}=\frac{\alpha \beta}{(\alpha+\beta+1)(\alpha+\beta)^{2}}\) . (b) Show that when \(\alpha>1\) and \(\beta>1\), the mode is at \(x=(\alpha-1) /(\alpha+\beta-2)\). Equation Transcription: Text Transcription: X Alpha Beta mu =alpha/alpha + beta sigma^2= alpha beta /(alpha + beta +1)(alpha+ beta)^2 alpha>1 Beta >1 x=(alpha-1)/(beta+-2)

When \(\alpha\) and \(\beta\) are integers and \(0<p<1\), we have \(\int_{0}^{p}\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}y^{\alpha-1}(1-y)^{\beta-1} d y=\sum_{y=\alpha}^{n}\left(\begin{array}{l}n \\y\end{array}\right) p^{y}(1-p)^{n-y}\) where \(n=\alpha+\beta-1\). Verify this formula when \(\alpha=4\) and \(\beta=3\). Hint: Integrate the left member by parts several times. Equation Transcription: Text Transcription: Alpha Beta 0<p<1 int_0^p gamma (alpha +beta)/gamma(alpha)(beta)y^alpha-1(1-y)^beta-1 dy=sum_y=alpha^n (n y) p^y(1-p)^n-y n=alpha +beta -1 alpha=4 beta=3

Evaluate \(\int_{0}^{0.4} \frac{\Gamma(7)}{\Gamma(4) \Gamma(3)} y^{3}(1-y)^{2} \ d y\) (a) Using integration. (b) Using the result of Exercise 5.2-10. Equation Transcription: Text Transcription: int_0^0.4 gamma (7)/gamma(4)gamma(3)y^3(1-y)^2 dy

Let \(W_{1}, W_{2}\) be independent, each with a Cauchy distribution. In this exercise we find the pdf of the sample mean, $\left(W_{1}+W_{2}\right) / 2\). (a) Show that the pdf of \(X_{1}=(1 / 2) W_{1}\) is \(f\left(x_{1}\right)=\frac{2}{\pi\left(1+4 x_{1}^{2}\right)}, \quad-\infty<x_{1}<\infty\) (b) Let \(Y_{1}=X_{1}+X_{2}=\bar{W}\) and \(Y_{2}=X_{1}\), where \(X_{2}=(1 / 2) W_{2}\). Show that the joint pdf of \(Y_{1}\) and \(Y_{2}\) is \(g\left(y_{1}, y_{2}\right)=f\left(y_{1}-y_{2}\right) f\left(y_{2}\right), \quad-\infty<y_{1}<\infty\), \(-\infty<y_{2}<\infty\) (c) Show that the pdf of \(Y_{1}=\bar{W}\) is given by the convolution formula, \(g_{1}\left(y_{1}\right)=\int_{-\infty}^{\infty} f\left(y_{1}-y_{2}\right) f\left(y_{2}\right) d y_{2}\). (d) Show that \(g_{1}\left(y_{1}\right)=\frac{1}{\pi\left(1+y_{1}^{2}\right)}, \quad-\infty<y_{1}<\infty\) That is, the pdf of \(\bar{W}\) is the same as that of an individual \(W\) Equation Transcription: . Text Transcription: W_1,W_2 (W_1+W_2)/2 X_1=(1/2)W_1 f(x_1)=2/pi(1+4x_1^2), -infinity <x1< infinity Y_1=X_1+X_2= bar W Y_2=X_1 X_2=(1/2)W_2. Y_1 Y_2 g(y_1,y_2=f(y_1-y_2)f(y_2), - infinity <y_1< -infinity<y2<infinity Y_1=bar W g_1(y_1)= int_-infinity^infinity (fy_1-y_2)f(y_2)dy_2 g_1(y_1)=1/pi(1+y_1^2), -infinity< y_1<infinity Bar W W

Let \(X_{1}\) and \(X_{2}\) have independent gamma distributions with parameters \(\alpha, \theta\) and \(\beta, \theta\), respectively. Let \(W=X_{1} /\left(X_{1}+X_{2}\right)\). Use a method similar to that given in the derivation of the \(F\) distribution (Example 5.2-4) to show that the pdf of \(W\) is \(g(w)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} w^{\alpha-1}(1-w)^{\beta-1}, \quad 0<w<1\). We say that W has a beta distribution with parameters \(\alpha\) and \(\beta\). (See Example 5.2-3.) Equation Transcription: Text Transcription: X_1 X_2 Alpha, theta Beta, theta W=X_1/(X_1+X_2) F W g(w)=gamma(alpha+beta)/gamma (alpha) gamma(beta)w^-1(1-w)^ beta-1, 0<w<1 Alpha beta

Problem 13E Let X1,X2 be independent random variables representing lifetimes (in hours) of two key components of a device that fails when and only when both components fail. Say each Xi has an exponential distribution with mean 1000. Let Y1 = min(X1,X2) and Y2 = max(X1,X2), so that the space of Y1,Y2 is 0 < y1 < y2 < ?. (a) Find G(y1, y2) = P(Y1 ? y1,Y2 ? y2). (b) Compute the probability that the device fails after 1200 hours; that is, compute P(Y2 > 1200).

A company provides earthquake insurance. The premium \(X\) is modeled by the pdf \(f(x)=\frac{x}{5^{2}} e^{-x / 5}, \quad 0<x<\infty\), while the claims \(Y\) have the pdf \(g(y)=\frac{1}{5} e^{-y / 5}, \quad 0<y<\infty\) . If \(X\) and \(Y\) are independent, find the pdf of \(Z=X / Y\). Equation Transcription: Text Transcription: f(x)=x/5^2 e^-x/5, 0<x<infinity, g(y)=? e^-y/5, 0<y<infinity. X Y Z=X/Y

Let \(X_1, X_2\) denote two independent random variables, each with a \(\chi^2(2)\) distribution. Find the joint pdf of \(Y_1 = X_1\) and \(Y_2 = X_2 + X_1\). Note that the support of \(Y_1, Y_2\) is \(0 < y_1 < y_2 < \infty\). Also, find the marginal pdf of each of \(Y_1\) and \(Y_2\). Are \(Y_1\) and \(Y_2\) independent?

Let X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Let Y1 = (X1/r1)/(X2/r2) and Y2 = X2. (a) Find the joint pdf of Y1 and Y2. (b) Determine the marginal pdf of Y1 and show that Y1 has an F distribution. (This is another, but equivalent, way of finding the pdf of F.)

Find the mean and the variance of an F random variable with r1 and r2 degrees of freedom by first finding E(U), E(1/V), E(U2), and E(1/V2).

Let the distribution of W be F(9, 24). Find the following: (a) F0.05(9, 24). (b) F0.95(9, 24). (c) P(0.277 W 2.70).

Let the distribution of W be F(8, 4). Find the following: (a) F0.01(8, 4). (b) F0.99(8, 4). (c) P(0.198 W 8.98).

Let X1 and X2 have independent gamma distributions with parameters , and , , respectively. Let W = X1/(X1 + X2). Use a method similar to that given in the derivation of the F distribution (Example 5.2-4) to show that the pdf of W is g(w) = ( + ) ()() w1(1 w) 1, 0 < w < 1.We say that W has a beta distribution with parameters and . (See Example 5.2-3.)

Let X1 and X2 be independent chi-square random variables with r1 and r2 degrees of freedom, respectively. Show that (a) U = X1/(X1 + X2) has a beta distribution with = r1/2 and = r2/2. (b) V = X2/(X1 + X2) has a beta distribution with = r2/2 and = r1/2.

Let X have a beta distribution with parameters and . (See Example 5.2-3.) (a) Show that the mean and variance of X are, respectively, = + and 2 = ( + + 1)( + )2 . (b) Show that when > 1 and > 1, the mode is at x = ( 1)/( + 2).

Determine the constant c such that f(x) = cx3(1 x)6, 0 < x < 1, is a pdf.

When \(\alpha\) and \(\beta\) are integers and 0 < p < 1, we have \(\int_{0}^{p} \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha) \Gamma(\beta)} y^{\alpha-1}(1-y)^{\beta-1} d y=\sum_{y=\alpha}^{n}\left(\begin{array}{l} n \\ y \end{array}\right) p^{y}(1-p)^{n-y} \), where \(n = \alpha + \beta - 1\). Verify this formula when \(\alpha = 4\) and \(\beta = 3\). Hint: Integrate the left member by parts several times.

Evaluate \(\int_{0}^{0.4} \frac{\Gamma(7)}{\Gamma(4) \Gamma(3)} y^{3}(1-y)^{2} d y\) (a) Using integration. (b) Using the result of Exercise 5.2-10.

Let W1, W2 be independent, each with a Cauchy distribution. In this exercise we find the pdf of the sample mean, (W1 + W2)/2. (a) Show that the pdf of X1 = (1/2)W1 is f(x1) = 2 1 + 4x2 1 , < x1 < . (b) Let Y1 = X1 + X2 = W and Y2 = X1, where X2 = (1/2)W2. Show that the joint pdf of Y1 and Y2 is g(y1, y2) = f(y1 y2)f(y2), < y1 < , < y2 < . (c) Show that the pdf of Y1 = W is given by the convolution formula, g1(y1) =  f(y1 y2)f(y2) dy2. (d) Show that g1(y1) = 1 1 + y2 1 , < y1 < . That is, the pdf of W is the same as that of an individual W. 5.

Let \(X_1, X_2\) be independent random variables representing lifetimes (in hours) of two key components of a device that fails when and only when both components fail. Say each \(X_i\) has an exponential distribution with mean 1000. Let \(Y_1 = \operatorname {min}(X_1, X_2)\) and \(Y_2 =\operatorname {max}(X_1, X_2)\), so that the space of \(Y_1, Y_2\) is \(0 < y_1 < y_2 < \infty\). (a) Find \(G(y_1, y_2) = P(Y_1 \le y_1, Y_2 \le y_2)\). (b) Compute the probability that the device fails after 1200 hours; that is, compute \(P(Y_2 > 1200)\).

A company provides earthquake insurance. The premium X is modeled by the pdf f(x) = x 52 ex/5, 0 < x < , while the claims Y have the pdf g( y) = 1 5 ey/5, 0 < y < . If X and Y are independent, find the pdf of Z = X/Y.

In Example 5.2-6, verify that the given transformation maps {(x1, x2):0 < x1 < 1, 0 < x2 < 1} onto {(z1, z2) : < z1 < , < z2 < }, except for a set of points that has probability 0. Hint: What is the image of vertical line segments? What is the image of horizontal line segments?

Let W have an F distribution with parameters \(r_1\) and \(r_2\). Show that \(Z = 1/[1 + (r_1/r_2)W]\) has a beta distribution.