91Ó°ÊÓ

In Problem \(6.3,\) calculate the conditional probability mass function of \(Y_{1}\) given that (a) \(Y_{2}=1\) (b) \(Y_{2}=0\)

An insurance company supposes that each person has an accident parameter and that the yearly number of accidents of someone whose accident parameter is \(\lambda\) is Poisson distributed with mean \(\lambda .\) They also suppose that the parameter value of a newly insured person can be assumed to be the value of a gamma random variable with parameters \(s\) and \(\alpha .\) If a newly insured person has \(n\) accidents in her first year, find the conditional density of her accident parameter. Also, determine the expected number of accidents that she will have in the following year.

Consider a sample of size 5 from a uniform distribution over \((0,1) .\) Compute the probability that the median is in the interval \(\left(\frac{1}{4}, \frac{3}{4}\right).\)

Let \(Z_{1}\) and \(Z_{2}\) be independent standard normal random variables. Show that \(X, Y\) has a bivariate normal distribution when \(X=Z_{1}, Y=Z_{1}+Z_{2}.\)

If \(X\) and \(Y\) are independent random variables both uniformly distributed over \((0,1),\) find the joint density function of \(R=\sqrt{X^{2}+Y^{2}}, \Theta=\tan ^{-1} Y / X.\)

Consider an urn containing \(n\) balls numbered \(1, \ldots, n,\) and suppose that \(k\) of them are randomly withdrawn.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 \(X_{1}, \ldots, X_{n}\) are exchangeable.