91Ó°ÊÓ

Let \(T\) have a \(t\) -distribution with 10 degrees of freedom. Find \(P(|T|>2.228)\) from either Table IV or, if available, \(\mathrm{R}\) or S-PLUS.

The mgf of a random variable \(X\) is \(\left(\frac{2}{3}+\frac{1}{3} e^{t}\right)^{9} .\) Show that $$ P(\mu-2 \sigma

. Let \(X\) and \(Y\) have a bivariate normal distribution with parameters \(\mu_{1}=\) \(5, \mu_{2}=10, \sigma_{1}^{2}=1, \sigma_{2}^{2}=25\), and \(\rho>0 .\) If \(P(4

Show that the failure rate (hazard function) of the Pareto distribution is $$ \frac{h(x)}{1-H(x)}=\frac{\alpha}{\beta^{-1}+x} $$ Find the failure rate (hazard function) of the Burr distribution with cdf $$ G(y)=1-\left(\frac{1}{1+\beta y^{\tau}}\right)^{\alpha}, \quad 0 \leq y<\infty . $$ In each of these two failure rates, note what happens as the value of the variable increases.

Let \(X\) have a gamma distribution with pdf $$ f(x)=\frac{1}{\beta^{2}} x e^{-x / \beta}, \quad 0

Compute the measures of skewness and kurtosis of a gamma distribution which has parameters \(\alpha\) and \(\beta\).

Determine the \(90 t h\) percentile of the distribution, which is \(N(65,25)\).

. Let \(X\) and \(Y\) have a bivariate normal distribution with parameters \(\mu_{1}=\) \(\mu_{2}=0, \sigma_{1}^{2}=\sigma_{2}^{2}=1\), and correlation coefficient \(\rho .\) Find the distribution of the random variable \(Z=a X+b Y\) in which \(a\) and \(b\) are nonzero constants.

. Using the computer, obtain plots of the pdfs of chi-squared distributions with degrees of freedom \(r=1,2,5,10,20\). Comment on the plots.

Let \(X\) have a Poisson distribution with mean \(1 .\) Compute, if it exists, the expected value \(E(X !)\)