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Chapter 1: Problem 7
Let \(X\) have a pmf \(p(x)=\frac{1}{3}, x=1,2,3\), zero elsewhere. Find the pmf of \(Y=2 X+1\)
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Let \(X\) have the cdf \(F(x)\) that is a mixture of the continuous and discrete types, namely $$F(x)=\left\\{\begin{array}{ll}0 & x<0 \\\\\frac{x+1}{4} & 0 \leq x<1 \\ 1 & 1 \leq x\end{array}\right.$$ Determine reasonable definitions of \(\mu=E(X)\) and \(\sigma^{2}=\operatorname{var}(X)\) and compute each.
Generalize Exercise \(1.2 .5\) to obtain $$\left(C_{1} \cup C_{2} \cup \cdots \cup C_{k}\right)^{c}=C_{1}^{c} \cap C_{2}^{c} \cap \cdots \cap C_{k}^{c}$$ Say that \(C_{1}, C_{2}, \ldots, C_{k}\) are independent events that have respective probabilities \(p_{1}, p_{2}, \ldots, p_{k} .\) Argue that the probability of at least one of \(C_{1}, C_{2}, \ldots, C_{k}\) is equal to $$1-\left(1-p_{1}\right)\left(1-p_{2}\right) \cdots\left(1-p_{k}\right)$$
Let \(X\) have the pdf \(f(x)=\frac{1}{x^{2}}, 1
Let 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 \(X\) be a random variable such that \(R(t)=E\left(e^{t(X-b)}\right)\) exists
for \(t\) such that \(-h
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