91Ó°ÊÓ

Suppose that 3 balls are chosen without replacement from an urn consisting of 5 white and 8 red balls. Let \(X_{i}\) equal 1 if the \(i\) th ball selected is white, and let it equal 0 otherwise. Give the joint probability mass function of (a) \(X_{1}, X_{2}\) (b) \(X_{1}, X_{2}, X_{3}\).

Consider a sequence of independent Bernoulli trials, each of which is a success with probability \(p\) Let \(X_{1}\) be the number of failures preceding the first success, and let \(X_{2}\) be the number of failures between the first two successes. Find the joint mass function of \(X_{1}\) and \(X_{2}\).

The number of people that enter a drugstore in a given hour is a Poisson random variable with parameter \(\lambda=10 .\) Compute the conditional probability that at most 3 men entered the drugstore, given that 10 women entered in that hour. What assumptions have you made?

The random vector \((X, Y)\) is said to be uniformly distributed over a region \(R\) in the plane if, for some constant \(c,\) its joint density is $$f(x, y)=\left\\{\begin{array}{ll}c & \text { if }(x, y) \in R \\\0 & \text { otherwise }\end{array}\right.$$ (a) Show that \(1 / c=\) area of region \(R\) Suppose that \((X, Y)\) is uniformly distributed over the square centered at (0,0) and with sides of length 2. (b) Show that \(X\) and \(Y\) are independent, with each being distributed uniformly over (-1,1) (c) What is the probability that \((X, Y)\) lies in the circle of radius 1 centered at the origin? That is, find \(P\left\\{X^{2}+Y^{2} \leq 1\right\\}\).

Show that \(f(x, y)=1 / x, 0

Suppose that \(A, B, C,\) are independent random variables, each being uniformly distributed over (0,1) (a) What is the joint cumulative distribution function of \(A, B, C ?\) (b) What is the probability that all of the roots of the equation \(A x^{2}+B x+C=0\) are real?

Jill's bowling scores are approximately normally distributed with mean 170 and standard deviation \(20,\) while Jack's scores are approximately normally distributed with mean 160 and standard deviation 15\. If Jack and Jill each bowl one game, then assuming that their scores are independent random variables, approximate the probability that (a) Jack's score is higher; (b) the total of their scores is above \(350 .\)

In Problem \(5,\) calculate the conditional probability mass function of \(Y_{1}\) given that (a) \(Y_{2}=1\) (b) \(Y_{2}=0\)

The joint density function of \(X\) and \(Y\) is given by \(f(x, y)=x e^{-x(y+1)} \quad x>0, y>0\) (a) Find the conditional density of \(X,\) given \(Y=y\) and that of \(Y,\) given \(X=x\) (b) Find the density function of \(Z=X Y\).

The joint density of \(X\) and \(Y\) is \(f(x, y)=c\left(x^{2}-y^{2}\right) e^{-x} \quad 0 \leq x<\infty,-x \leq y \leq x\) Find the conditional distribution of \(Y,\) given \(X=x\).