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Chapter 3: Problem 26
Compute the measures of skewness and kurtosis of the Poisson distribution with mean \(\mu\).
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Let the stochastically independent random variables \(X_{1}\) and \(X_{2}\) have binomial distributions with parameters \(n_{1}=3, p_{1}=\frac{2}{3}\) and \(n_{2}=4, p_{2}=\frac{1}{2}\) respectively. Compute \(\operatorname{Pr}\left(X_{1}=X_{2}\right) .\) Hint. List the four mutually exclusive ways that \(X_{1}=\dot{X}_{2}\) and compute the probability of each.
Let \(X\) be \(n\left(\mu, \sigma^{2}\right)\) so that \(\operatorname{Pr}(X<89)=0.90\) and \(\operatorname{Pr}(X<94)=\) 0.95. Find \(\mu\) and \(\sigma^{2}\).
Let \(X\) and \(Y\) be stochastically independent random variables, each with a distribution that is \(n(0,1)\). Let \(Z=X+Y\). Find the integral that represents the distribution function \(G(z)=\operatorname{Pr}(X+Y \leq z)\) of \(Z .\) Determine the p.d.f. of \(Z .\) Hint. We have that \(G(z)=\int_{-\infty}^{\infty} H(x, z) d x\), where $$H(x, z)=\int_{-\infty}^{z-x} \frac{1}{2 \pi} \exp \left[-\left(x^{2}+y^{2}\right) / 2\right] d y$$ Find \(G^{\prime}(z)\) by evaluating \(\int_{-\infty}^{\infty}[\partial H(x, z) / \partial z] d x\).
Show that the graph of a p.d.f. \(n\left(\mu, \sigma^{2}\right)\) has points of inflection at \(x=\mu-\sigma\) and \(x=\mu+\sigma\).
Let the random variable \(X\) have the p.d.f.
$$f(x)=\frac{2}{\sqrt{2 \pi}} e^{-x^{2} / 2}, \quad 0
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