91Ó°ÊÓ

Two fair dice are rolled. Find the joint probability mass function of \(X\) and \(Y\) when (a) \(X\) is the largest value obtained on any die and \(Y\) is the sum of the values; (b) \(X\) is the value on the first die and \(Y\) is the larger of the two values; (c) \(X\) is the smallest and \(Y\) is the largest value obtained on the dice.

If \(X\) and \(Y\) are independent continuous positive random variables, express the density function of (a) \(Z=X / Y\) and (b) \(Z=X Y\) in terms of the density functions of \(X\) and \(Y\). Evaluate these expressions in the special case where \(X\) and \(Y\) are both exponential random variables.

Show analytically (by induction) that \(X_{1}+\cdots+X_{n}\) has a negative binomial distribution when the \(X_{i}, i=1, \ldots, n\) are independent and identically distributed geometric random variables. Also, give a second argument that verifies the above without any need for computations.

The following dartboard is a square whose sides are of length 6 . The three circles are all centered at the center of the board and are of radii 1,2 , and 3. Darts landing within the circle of radius 1 score 30 points, those landing outside this circle but within the circle of radius 2 are worth 20 points, and those landing outside the circle of radius 2 but within the circle of radius 3 are worth 10 points. Darts that do not land within the circle of radius 3 do not score any points. Assuming that each dart that you throw will, independent of what occurred on your previous throws, land on a point uniformly distributed in the square, find the probabilities of the following events. (a) You score 20 on a throw of the dart. (b). You score at least 20 on a throw of the dart. (c) You score 0 on a throw of the dart. (d) The expected value of your score on a throw of the dart. (e) Both of your first two throws score at least 10 . (f) Your total score after two throws is 30 .

Two points are selected randomly on a line of length \(L\) so as to be on opposite sides of the midpoint of the line. [In other words, the two points \(X\) and \(Y\) are independent random variables such that \(X\) is uniformly distributed over (0, \(L / 2\) ) and \(Y\) is uniformly distributed over \((L / 2, L) .]\) Find the probability that the distance between the two points is greater than \(L / 3\).

The joint density of \(X\) and \(Y\) is given by $$ f(x, y)= \begin{cases}x e^{-(x+y)} & x>0, y>0 \\ 0 & \text { otherwise }\end{cases} $$ Are \(X\) and \(Y\) independent? What if \(f(x, y)\) were given by $$ f(x, y)= \begin{cases}2 & 0

The random variables \(X\) and \(Y\) are said to have a bivariate normal distribution if their joint density function is given by $$ \begin{aligned} f(x, y) &=\frac{1}{2 \pi \sigma_{x} \sigma_{y} \sqrt{1-\rho^{2}}} \\ \quad \times & \exp \left\\{-\frac{1}{2\left(1-\rho^{2}\right)}\left[\left(\frac{x-\mu_{x}}{\sigma_{x}}\right)^{2}+\left(\frac{y-\mu_{y}}{\sigma_{y}}\right)^{2}-2 \rho \frac{\left(x-\mu_{x}\right)\left(y-\mu_{y}\right)}{\sigma_{x} \sigma_{y}}\right]\right\\} \end{aligned} $$ (a) Show that the conditional density of \(X\), given that \(Y=y\), is the normal density with parameters $$ \mu_{x}+\rho \frac{\sigma_{x}}{\sigma_{y}}\left(y-\mu_{y}\right) \quad \text { and } \quad \sigma_{x}^{2}\left(1-\rho^{2}\right) $$ (b) Show that \(X\) and \(Y\) are both normal random variables with respective parameters \(\mu_{x}, \sigma_{x}^{2}\) and \(\mu_{y}, \sigma_{y}^{2}\) (c) Show that \(X\) and \(Y\) are independent when \(\rho=0\).

According to the U.S. National Center for Health Statistics, \(25.2\) percent of males and \(23.6\) percent of females never eat breakfast. Suppose that random samples of 200 men and 200 women are chosen. Approximate the probability that (a) at least 110 of these 400 people never eat breakfast; (b) the number of the women who never eat breakfast is at least as large as the number of the men who never eat breakfast.

Choose a number \(X\) at random from the set of numbers \(\\{1,2,3,4,5\\}\). Now choose a number at random from the subset no larger than \(X\), that is, from \(\\{1, \ldots, X]\). Call this second number \(Y\). (a) Find the joint mass function of \(X\) and \(Y\). (b) Find the conditional mass function of \(X\) given that \(Y=i\). Do it for \(i=1,2,3,4,5 .\) (c) Are \(X\) and \(Y\) independent? Why?

Two dice are rolled. Let \(X\) and \(Y\) denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of \(Y\) given \(X=i\), for \(i=1,2, \ldots, 6 .\) Are \(X\) and \(Y\) independent? Why?