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Chapter 1: Problem 11
For every one-dimensional set \(C\) for which the integral exists, let \(Q(C)=\)
\(\int_{C} f(x) d x\), where \(f(x)=6 x(1-x), 0
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From a bowl containing 5 red, 3 white, and 7 blue chips, select 4 at random and without replacement. Compute the conditional probability of 1 red, 0 white, and 3 blue chips, given that there are at least 3 blue chips in this sample of 4 chips.
Let \(X\) have the exponential pdf, \(f(x)=\beta^{-1} \exp \\{-x / \beta\\},
0
Let a bowl contain 10 chips of the same size and shape. One and only one of these chips is red. Continue to draw chips from the bowl, one at a time and at random and without replacement, until the red chip is drawn. (a) Find the pmf of \(X\), the number of trials needed to draw the red chip. (b) Compute \(P(X \leq 4)\).
Hunters \(\mathrm{A}\) and \(\mathrm{B}\) shoot at a target; the probabilities of hitting the target are \(p_{1}\) and \(p_{2}\), respectively. Assuming independence, can \(p_{1}\) and \(p_{2}\) be selected so that \(P(\) zero hits \()=P(\) one hit \()=P(\) two hits \() ?\)
Let \(X\) be a random variable with mean \(\mu\) and let \(E\left[(X-\mu)^{2 k}\right]\) exist. Show, with \(d>0\), that \(P(|X-\mu| \geq d) \leq E\left[(X-\mu)^{2 k}\right] / d^{2 k}\). This is essentially Chebyshev's inequality when \(k=1\). The fact that this holds for all \(k=1,2,3, \ldots\), when those \((2 k)\) th moments exist, usually provides a much smaller upper bound for \(P(|X-\mu| \geq d)\) than does Chebyshev's result.
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