Chapter 1: Problem 9
Let \(X\) denote a random variable for which \(E\left[(X-a)^{2}\right]\) exists. Give an example of a distribution of a discrete type such that this expectation is zero. Such a distribution is called a degenerate distribution.
Chapter 1: Problem 9
Let \(X\) denote a random variable for which \(E\left[(X-a)^{2}\right]\) exists. Give an example of a distribution of a discrete type such that this expectation is zero. Such a distribution is called a degenerate distribution.
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Get started for freeLet \(X\) 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.
Let \(X\) equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of \(X\) and compute the probability that \(X\) is equal to an odd number.
(Monte Hall Problem). Suppose there are three curtains. Behind one curtain there is a nice prize while behind the other two there are worthless prizes. A contestant selects one curtain at random, and then Monte Hall opens one of the other two curtains to reveal a worthless prize. Hall then expresses the willingness to trade the curtain that the contestant has chosen for the other curtain that has not been opened. Should the contestant switch curtains or stick with the one that she has? If she sticks with the curtain she has then the probability of winning the prize is \(1 / 3 .\) Hence, to answer the question determine the probability that she wins the prize if she switches.
Let \(X\) be a random variable with a pdf \(f(x)\) and mgf \(M(t)\). Suppose \(f\) is symmetric about \(0,(f(-x)=f(x))\). Show that \(M(-t)=M(t)\).
Given \(\int_{C}\left[1 / \pi\left(1+x^{2}\right)\right] d x\), where \(C \subset
\mathcal{C}=\\{x:-\infty
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