Chapter 1: Problem 25
Let \(X\) have the exponential pdf, \(f(x)=\beta^{-1} \exp \\{-x / \beta\\},
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Chapter 1: Problem 25
Let \(X\) have the exponential pdf, \(f(x)=\beta^{-1} \exp \\{-x / \beta\\},
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Get started for freeLet the random variable \(X\) have pmf $$p(x)=\left\\{\begin{array}{ll}p & x=-1,1 \\\1-2 p & x=0 \\\0 & \text { elsewhere }\end{array}\right.$$ where \(0
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.
Suppose that \(p(x)=\frac{1}{5}, x=1,2,3,4,5\), zero elsewhere, is the pmf of the discrete type random variable \(X\). Compute \(E(X)\) and \(E\left(X^{2}\right)\). Use these two results to find \(E\left[(X+2)^{2}\right]\) by writing \((X+2)^{2}=X^{2}+4 X+4\)
If \(C_{1}\) and \(C_{2}\) are independent events, show that the following pairs of events are also independent: (a) \(C_{1}\) and \(C_{2}^{c}\), (b) \(C_{1}^{c}\) and \(C_{2}\), and (c) \(C_{1}^{c}\) and \(C_{2}^{c}\). Hint: In (a), write \(P\left(C_{1} \cap C_{2}^{c}\right)=P\left(C_{1}\right) P\left(C_{2}^{c} \mid C_{1}\right)=P\left(C_{1}\right)\left[1-P\left(C_{2} \mid C_{1}\right)\right]\). From independence of \(C_{1}\) and \(C_{2}, P\left(C_{2} \mid C_{1}\right)=P\left(C_{2}\right)\).
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 \(P(X \leq 4)\).
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