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Chapter 3: Problem 9
Compute the measures of skewness and kurtosis of the Poisson distribution with mean \(\mu\).
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If \(X\) is \(N(75,100)\), find \(P(X<60)\) and \(P(70
Let \(X\) be a random variable such that \(E\left(X^{m}\right)=(m+1) ! 2^{m}, m=1,2,3, \ldots\). Determine the mgf and the distribution of \(X\).
Let \(X\) and \(Y\) have a bivariate normal distribution with parameters
\(\mu_{1}=\) 3, \(\mu_{2}=1, \sigma_{1}^{2}=16, \sigma_{2}^{2}=25\), and
\(\rho=\frac{3}{5} .\) Determine the following probabilities:
(a) \(P(3
. Let \(X\) have a Poisson distribution with parameter \(m .\) If \(m\) is an experimental value of a random variable having a gamma distribution with \(\alpha=2\) and \(\beta=1\), compute \(P(X=0,1,2)\). Hint: Find an expression that represents the joint distribution of \(X\) and \(m\). Then integrate out \(m\) to find the marginal distribution of \(X\).
. Let \(X_{1}, X_{2}\), and \(X_{3}\) be three independent chi-square variables with \(r_{1}, r_{2}\), and \(r_{3}\) degrees of freedom, respectively. (a) Show that \(Y_{1}=X_{1} / X_{2}\) and \(Y_{2}=X_{1}+X_{2}\) are independent and that \(Y_{2}\) is \(\chi^{2}\left(r_{1}+r_{2}\right)\) (b) Deduce that $$ \frac{X_{1} / r_{1}}{X_{2} / r_{2}} \quad \text { and } \quad \frac{X_{3} / r_{3}}{\left(X_{1}+X_{2}\right) /\left(r_{1}+r_{2}\right)} $$ are independent \(F\) -variables.
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