Chapter 3: Problem 12
Compute the measures of skewness and kurtosis of the Poisson distribution with mean \(\mu\).
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Suppose \(X\) is a random variable with the pdf \(f(x)\) which is symmetric about \(0 ;\) i.e., \(f(-x)=f(x) .\) Show that \(F(-x)=1-F(x)\), for all \(x\) in the support of \(X\).
Let \(X\) and \(Y\) have a bivariate normal distribution with parameters
\(\mu_{1}=\) \(5, \mu_{2}=10, \sigma_{1}^{2}=1, \sigma_{2}^{2}=25\), and \(\rho>0
.\) If \(P(4
Continuing with Exercise \(3.2 .8\), make a page of four overlay plots for the following 4 Poisson and binomial combinations: \(\lambda=2, p=0.02 ; \lambda=10, p=0.10\); \(\lambda=30, p=0.30 ; \lambda=50, p=0.50 .\) Use \(n=100\) in each situation. Plot the subset of the binomial range that is between \(n p \pm \sqrt{n p(1-p)} .\) For each situation, comment on the goodness of the Poisson approximation to the binomial.
Three fair dice are cast. In 10 independent casts, let \(X\) be the number of times all three faces are alike and let \(Y\) be the number of times only two faces are alike. Find the joint \(\mathrm{pmf}\) of \(X\) and \(Y\) and compute \(E(6 X Y)\).
Let the random variable \(X\) have a distribution that is \(N\left(\mu, \sigma^{2}\right)\). (a) Does the random variable \(Y=X^{2}\) also have a normal distribution? (b) Would the random variable \(Y=a X+b, a\) and \(b\) nonzero constants have a normal distribution? Hint: \(\operatorname{In}\) each case, first determine \(P(Y \leq y)\).
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