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A bin of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N1 the number of tests made until the first defective is identified and by N2 the number of additional tests until the second defective is identified. Find the joint probability mass function of N1 and N2.

In Example $8$b, let $Y_{k+1}=n+1-\underset{i=1}{\overset{k}{鈭�}}{Y}_i$Show that $Y_1,....,Y_k,Y_{k+1}$ are exchangeable. Note that $Y_{k+1}$ is the number of balls one must observe to obtain a special ball if one considers the balls in their reverse order of withdrawal.

Consider an urn containing n balls numbered $1,.....,n$and suppose that k of them are randomly withdrawn. Let $X_i$equal $1$if ball number $i$is removed and let $X_i$be $0$ otherwise. Show that $X_1,......X_n$ are exchangeable .

1. If X has a gamma distribution with parameters $(n,饾浑)$what is the distribution of$cX,c>0$
2. Show that $\frac{{\mathrm{饾挸}}_{2n}^2}{2饾浑}$has a gamma distribution with parameters$(n,饾潃)$ when n is a positive integer and ${\mathrm{饾挸}}_{2n}^2$is a chi-squared random variable with $2n$degrees of freedom

(a) If X has a gamma distribution with parameters (t, 位), what is the distribution of cX, c > 0?

(b) Show that $\frac{{\mathrm{饾挸}}_{2n}^2}{2饾潃}$has a gamma distribution with parameters n and 位when nis a

positive integer and ${\mathrm{饾挸}}_{2n}^2$is a chi-squared random variable with 2n degrees of freedom.

The joint density function of X and Y is $f(x,y)=\left\{\begin{array}{l}xy鈥呪赌呪赌呪赌�0<x<1,0<y<2\\ 0鈥呪赌呪赌呪赌�\text{otherwise}\end{array}\right.$

(a) Are X and Y independent?

(b) Find the density function of X.

(c) Find the density function of Y.

(d) Find the joint distribution function.

(e) Find$E\left[Y\right].$

(f) Find$P[X+Y<1]$

Consider a sequence of independent Bernoulli trials, each of which is a success with probability p. Let X1 be the number of failures preceding the first success, and let X2 be the number of failures between the first two successes. Find the joint mass function of X1 and X2.

(a) If X has a gamma distribution with parameters$(n,饾浑)$what is the distribution of $cX,c>0$

(b) Show that $\frac{{\mathrm{饾挸}}_{2n}^2}{2饾浑}$has a gamma distribution with parameters $(n,饾潃)$when n is a positive integer and ${\mathrm{饾挸}}_{2n}^2$is a chi-squared random variable with $2n$degrees of freedom

The joint probability density function of X and Y is given by

f(x, y) = c(y² 鈭� x²)e^-y 鈭抷 鈥� x 鈥� y, 0 < y < q .

(a) Find c.

(b) Find the marginal densities of X and Y.

(c) Find E[X].

Let X and Y be independent continuous random variables with respective hazard rate functions 位X(t) and 位Y(t), and set W = min(X, Y).

(a) Determine the distribution function of W in terms of those of X and Y.

(b) Show that 位W(t), the hazard rate function of W, is given by 位W(t) = 位X(t) + 位Y(t)