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 the number \(X\) of accidents have a Poisson distribution with mean \(\lambda \theta\). Suppose \(\lambda\), the liability to have an accident, has, given \(\theta\), a gamma pdf with parameters \(\alpha=h\) and \(\beta=h^{-1} ;\) and \(\theta\), an accident proneness factor, has a generalized Pareto pdf with parameters \(\alpha, \lambda=h\), and \(k\). Show that the unconditional pdf of \(X\) is $$ \frac{\Gamma(\alpha+k) \Gamma(\alpha+h) \Gamma(\alpha+h+k) \Gamma(h+k) \Gamma(k+x)}{\Gamma(\alpha) \Gamma(\alpha+k+h) \Gamma(h) \Gamma(k) \Gamma(\alpha+h+k+x) x !}, \quad x=0,1,2, \ldots $$ sometimes called the generalized Waring pdf.
. Suppose \(X_{1}, X_{2}\) are iid with a common standard normal distribution.
Find the joint pdf of \(Y_{1}=X_{1}^{2}+X_{2}^{2}\) and \(Y_{2}=X_{2}\) and the
marginal pdf of \(Y_{1}\). Hint: \(\quad\) Note that the space of \(Y_{1}\) and
\(Y_{2}\) is given by \(-\sqrt{y_{1}}
Let the \(\operatorname{pmf} p(x)\) be positive on and only on the nonnegative integers. Given that \(p(x)=(4 / x) p(x-1), x=1,2,3, \ldots\) Find \(p(x)\). Hint: \(\quad\) Note that \(p(1)=4 p(0), p(2)=\left(4^{2} / 2 !\right) p(0)\), and so on. That is, find each \(p(x)\) in terms of \(p(0)\) and then determine \(p(0)\) from $$ 1=p(0)+p(1)+p(2)+\cdots $$
Let \(X\) be \(N(5,10)\). Find \(P\left[0.04<(X-5)^{2}<38.4\right]\).
. Let \(T\) have a \(t\) -distribution with 14 degrees of freedom. Determine \(b\)
so that \(P(-b
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