Chapter 3: Problem 9
Toss two nickels and three dimes at random. Make appropriate assumptions and compute the probability that there are more heads showing on the nickels than on the dimes.
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Let \(X\) and \(Y\) be independent random variables, each with a distribution that is \(N(0,1) .\) Let \(Z=X+Y .\) Find the integral that represents the cdf \(G(z)=\) \(P(X+Y \leq z)\) of \(Z .\) Determine the pdf 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 \(\int_{-\infty}^{\infty}[\partial H(x, z) / \partial z] d x\).
. Let \(X\) and \(Y\) have a bivariate normal distribution with parameters \(\mu_{1}=\) \(\mu_{2}=0, \sigma_{1}^{2}=\sigma_{2}^{2}=1\), and correlation coefficient \(\rho .\) Find the distribution of the random variable \(Z=a X+b Y\) in which \(a\) and \(b\) are nonzero constants.
Let \(X\) have a Poisson distribution with \(\mu=100\). Use Chebyshev's inequality
to determine a lower bound for \(P(75
. Suppose \(X\) is distributed \(N_{n}(\boldsymbol{\mu}, \boldsymbol{\Sigma})\). 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 \(\mu\).
Show that the failure rate (hazard function) of the Pareto distribution is $$ \frac{h(x)}{1-H(x)}=\frac{\alpha}{\beta^{-1}+x} $$ Find the failure rate (hazard function) of the Burr distribution with cdf $$ G(y)=1-\left(\frac{1}{1+\beta y^{\tau}}\right)^{\alpha}, \quad 0 \leq y<\infty . $$ In each of these two failure rates, note what happens as the value of the variable increases.
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