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Chapter 1: Problem 3
If \(X\) is a random variable such that \(E(X)=3\) and \(E\left(X^{2}\right)=13\), use Chebyshev's inequality to determine a lower bound for the probability \(P(-2<\) \(X<8)\)
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Find the cdf \(F(x)\) associated with each of the following probability density
functions. Sketch the graphs of \(f(x)\) and \(F(x)\).
(a) \(f(x)=3(1-x)^{2}, 0
Find the moments of the distribution that has mgf \(M(t)=(1-t)^{-3}, t<1\). Hint: Find the Maclaurin series for \(M(t) .\)
Let \(X\) be a random variable with space \(\mathcal{D}\). For \(D \subset \mathcal{D}\), recall that the probability induced by \(X\) is \(P_{X}(D)=P[\\{c: X(c) \in D\\}] .\) Show that \(P_{X}(D)\) is a probability by showing the following: (a) \(P_{X}(\mathcal{D})=1\). (b) \(P_{X}(D) \geq 0\). (c) For a sequence of sets \(\left\\{D_{n}\right\\}\) in \(\mathcal{D}\), show that $$ \left\\{c: X(c) \in \cup_{n} D_{n}\right\\}=\cup_{n}\left\\{c: X(c) \in D_{n}\right\\} $$ (d) Use part (c) to show that if \(\left\\{D_{n}\right\\}\) is sequence of mutually exclusive events, then $$ P_{X}\left(\cup_{n=1}^{\infty} D_{n}\right)=\sum_{n=1}^{\infty} P\left(D_{n}\right) $$
After a hard-fought football game, it was reported that, of the 11 starting players, 8 hurt a hip, 6 hurt an arm, 5 hurt a knee, 3 hurt both a hip and an arm, 2 hurt both a hip and a knee, 1 hurt both an arm and a knee, and no one hurt all three. Comment on the accuracy of the report.
(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? To answer the question, determine the probability that she wins the prize if she switches.
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