Chapter 1: Problem 6
For each of the following pdfs of \(X\), find \(P(|X|<1)\) and
\(P\left(X^{2}<9\right)\).
(a) \(f(x)=x^{2} / 18,-3
Chapter 1: Problem 6
For each of the following pdfs of \(X\), find \(P(|X|<1)\) and
\(P\left(X^{2}<9\right)\).
(a) \(f(x)=x^{2} / 18,-3
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Let \(\psi(t)=\log M(t)\), where \(M(t)\) is the mgf of a distribution. Prove that \(\psi^{\prime}(0)=\mu\) and \(\psi^{\prime \prime}(0)=\sigma^{2}\). The function \(\psi(t)\) is called the cumulant generating function.
Let \(X\) have the pdf \(f(x)=(x+2) / 18,-2
Cast a die a number of independent times until a six appears on the up side of the die. (a) Find the pmf \(p(x)\) of \(X\), the number of casts needed to obtain that first six. (b) Show that \(\sum_{x=1}^{\infty} p(x)=1\). (c) Determine \(P(X=1,3,5,7, \ldots)\). (d) Find the cdf \(F(x)=P(X \leq x)\).
(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.
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