Chapter 3

Let \(T=W / \sqrt{V / r}\), where the independent variables \(W\) and \(V\) are, respectively, normal with mean zero and variance 1 and chi-square with \(r\) degrees of freedom. Show that \(T^{2}\) has an \(F\) -distribution with parameters \(r_{1}=1\) and \(r_{2}=r\). Hint: What is the distribution of the numerator of \(T^{2} ?\)

Let the number of chocolate chips in a certain type of cookie have a Poisson distribution. We want the probability that a cookie of this type contains at least two chocolate chips to be greater than \(0.99 .\) Find the smallest value of the mean that the distribution can take.

Let the number \(X\) of accidents have a Poisson distribution with mean \(\lambda \theta\). Suppose \(\lambda\), the liability to have an accident, has, given \(\theta\), a gamma pdf with parameters \(\alpha=h\) and \(\beta=h^{-1} ;\) and \(\theta\), an accident proneness factor, has a generalized Pareto pdf with parameters \(\alpha, \lambda=h\), and \(k .\) Show that the unconditional pdf of \(X\) is $$ \frac{\Gamma(\alpha+k) \Gamma(\alpha+h) \Gamma(\alpha+h+k) \Gamma(h+k) \Gamma(k+x)}{\Gamma(\alpha) \Gamma(\alpha+k+h) \Gamma(h) \Gamma(k) \Gamma(\alpha+h+k+x) x !}, \quad x=0,1,2, \ldots $$ sometimes called the generalized Waring pmf.

Let \(X, Y\), and \(Z\) have the joint pdf $$ \left(\frac{1}{2 \pi}\right)^{3 / 2} \exp \left(-\frac{x^{2}+y^{2}+z^{2}}{2}\right)\left[1+x y z \exp \left(-\frac{x^{2}+y^{2}+z^{2}}{2}\right)\right] $$ where \(-\infty

Let the random variable \(X\) have the pdf $$ f(x)=\frac{2}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad 0

Toss two nickels and three dimes at random. Make appropriate assumptions and compute the probability that there are more heads showing on the nickels than on the dimes.

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 conditional Burr distribution with fixed parameters \(\beta\) and \(\tau\), given parameter \(\alpha .\) (a) If \(\alpha\) has the geometric pmf \(p(1-p)^{\alpha}, \alpha=0,1,2, \ldots\), show that the unconditional distribution of \(X\) is a Burr distribution. (b) If \(\alpha\) has the exponential pdf \(\beta^{-1} e^{-\alpha / \beta}, \alpha>0\), find the unconditional pdf of \(X .\)

Show that the \(t\) -distribution with \(r=1\) degree of freedom and the Cauchy distribution are the same.

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.