Chapter 3: Problem 8
For the Burr distribution, show that $$ E\left(X^{k}\right)=\frac{1}{\beta^{k / \tau}} \Gamma\left(\alpha-\frac{k}{\tau}\right) \Gamma\left(\frac{k}{\tau}+1\right) / \Gamma(\alpha) $$ provided \(k<\alpha \tau\)
Chapter 3: Problem 8
For the Burr distribution, show that $$ E\left(X^{k}\right)=\frac{1}{\beta^{k / \tau}} \Gamma\left(\alpha-\frac{k}{\tau}\right) \Gamma\left(\frac{k}{\tau}+1\right) / \Gamma(\alpha) $$ provided \(k<\alpha \tau\)
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Get started for freeLet \(T=W / \sqrt{V / r}\), 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 the joint \(\operatorname{pmf} p(x, y)=e^{-2} /[x !(y-x) !], y=0,1,2, \ldots ;\) \(x=0,1, \ldots, y\), zero elsewhere. (a) Find the mgf \(M\left(t_{1}, t_{2}\right)\) of this joint distribution. (b) Compute the means, the variances, and the correlation coefficient of \(X\) and \(Y\). (c) Determine the conditional mean \(E(X \mid y)\). Hint: Note that $$ \sum_{x=0}^{y}\left[\exp \left(t_{1} x\right)\right] y ! /[x !(y-x) !]=\left[1+\exp \left(t_{1}\right)\right]^{y} $$ Why?
If \(X\) is \(\chi^{2}(5)\), determine the constants \(c\) and \(d\) so that
\(P(c
Let \(X\) have a gamma distribution with pdf
$$
f(x)=\frac{1}{\beta^{2}} x e^{-x / \beta}, \quad 0
. Let \(X\) be \(N(0,1)\). Use the moment-generating-function technique to show that \(Y=X^{2}\) is \(\chi^{2}(1)\) Hint: \(\quad\) Evaluate the integral that represents \(E\left(e^{t X^{2}}\right)\) by writing \(w=x \sqrt{1-2 t}\), \(t<\frac{1}{2}\)
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