91Ó°ÊÓ

Suppose that \(X\) and \(Y\) are continuous random variables with density functions \(f_{X}(x)\) and \(f_{Y}(y)\), respectively. Let \(f(x, y)\) denote the joint density function of \((X, Y)\). Show that $$ \int_{-\infty}^{\infty} f(x, y) d y=f_{X}(x) $$ and $$ \int_{-\infty}^{\infty} f(x, y) d x=f_{Y}(y) $$

Two cards are drawn from a bridge deck. What is the probability that the second card drawn is red?

The Polya urn model for contagion is as follows: We start with an urn which contains one white ball and one black ball. At each second we choose a ball at random from the urn and replace this ball and add one more of the color chosen. Write a program to simulate this model, and see if you can make any predictions about the proportion of white balls in the urn after a large number of draws. Is there a tendency to have a large fraction of balls of the same color in the long run?

A student is applying to Harvard and Dartmouth. He estimates that he has a probability of .5 of being accepted at Dartmouth and .3 of being accepted at Harvard. He further estimates the probability that he will be accepted by both is .2. What is the probability that he is accepted by Dartmouth if he is accepted by Harvard? Is the event "accepted at Harvard" independent of the event "accepted at Dartmouth"?

Four women, \(\mathrm{A}, \mathrm{B}, \mathrm{C},\) and \(\mathrm{D},\) check their hats, and the hats are returned in a random manner. Let \(\Omega\) be the set of all possible permutations of \(\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}\). Let \(X_{j}=1\) if the \(j\) th woman gets her own hat back and 0 otherwise. What is the distribution of \(X_{j} ?\) Are the \(X_{i}\) 's mutually independent?

A box has numbers from 1 to 10 . A number is drawn at random. Let \(X_{1}\) be the number drawn. This number is replaced, and the ten numbers mixed. A second number \(X_{2}\) is drawn. Find the distributions of \(X_{1}\) and \(X_{2}\). Are \(X_{1}\) and \(X_{2}\) independent? Answer the same questions if the first number is not replaced before the second is drawn.

A fair coin is tossed three times. Let \(X\) be the number of heads that turn up on the first two tosses and \(Y\) the number of heads that turn up on the third toss. Give the distribution of (a) the random variables \(X\) and \(Y\). (b) the random variable \(Z=X+Y\). (c) the random variable \(W=X-Y\).

Let \(\Omega\) be the sample space of an experiment. Let \(E\) be an event with \(P(E)>0\) and define \(m_{E}(\omega)\) by \(m_{E}(\omega)=m(\omega \mid E)\). Prove that \(m_{E}(\omega)\) is a distribution function on \(E,\) that is, that \(m_{E}(\omega) \geq 0\) and that \(\sum_{\omega \in \Omega} m_{E}(\omega)=1\). The function \(m_{E}\) is called the conditional distribution given \(E\).