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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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Compute the measures of skewness and kurtosis of a gamma distribution which has parameters \(\alpha\) and \(\beta\).
Consider a standard deck of 52 cards. Let \(X\) equal the number of aces in a sample of size 2 . (a) If the sampling is with replacement, obtain the pmf of \(X\). (b) If the sampling is without replacement, obtain the pmf of \(X\).
Show, for \(k=1,2, \ldots, n\), that $$\int_{p}^{1} \frac{n !}{(k-1) !(n-k) !} z^{k-1}(1-z)^{n-k} d z=\sum_{x=0}^{k-1}\left(\begin{array}{l}n \\\x\end{array}\right) p^{x}(1-p)^{n-x} .$$ This demonstrates the relationship between the cdfs of the \(\beta\) and binomial distributions.
Suppose \(\mathbf{X}\) is distributed \(N_{n}(\boldsymbol{\mu}, \mathbf{\Sigma}) .\) Let \(\bar{X}=n^{-1} \sum_{i=1}^{n} X_{i}\). (a) Write \(\bar{X}\) as aX for an appropriate vector a and apply Theorem \(3.5 .1\) to find the distribution of \(\bar{X}\). (b) Determine the distribution of \(\bar{X}\) if all of its component random variables \(X_{i}\) have the same mean \(\mu\).
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 \(X\).
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