91Ó°ÊÓ

Let \(X_{1}, X_{2}, \ldots, X_{k-1}\) have a multinomial distribution. (a) Find the mgf of \(X_{2}, X_{3}, \ldots, X_{k-1}\). (b) What is the pmf of \(X_{2}, X_{3}, \ldots, X_{k-1} ?\) (c) Determine the conditional pmf of \(X_{1}\) given that \(X_{2}=x_{2}, \ldots, X_{k-1}=x_{k-1}\). (d) What is the conditional expectation \(E\left(X_{1} \mid x_{2}, \ldots, x_{k-1}\right) ?\)

On the average, a grocer sells three of a certain article per week. How many of these should he have in stock so that the chance of his running out within a week is less than 0.01? Assume a Poisson distribution.

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 \(X\)

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 \(X\) 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.

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

Let \(X\) be \(b(2, p)\) and let \(Y\) be \(b(4, p)\). If \(P(X \geq 1)=\frac{5}{9}\), find \(P(Y \geq 1)\).

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. If \(P(X=1)=P(X=3)\), find the mode of the distribution.

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