Chapter 1: Problem 4
Let \(X\) be a random variable with mgf \(M(t),-h
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Chapter 1: Problem 4
Let \(X\) be a random variable with mgf \(M(t),-h
These are the key concepts you need to understand to accurately answer the question.
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Get started for freeEach of four persons fires one shot at a target. Let \(C_{k}\) denote the event that the target is hit by person \(k, k=1,2,3,4 .\) If \(C_{1}, C_{2}, C_{3}, C_{4}\) are independent and if \(P\left(C_{1}\right)=P\left(C_{2}\right)=0.7, P\left(C_{3}\right)=0.9\), and \(P\left(C_{4}\right)=0.4\), compute the probability that (a) all of them hit the target; (b) exactly one hits the target; (c) no one hits the target; (d) at least one hits the target.
Let the three mutually independent events \(C_{1}, C_{2}\), and \(C_{3}\) be such that \(P\left(C_{1}\right)=P\left(C_{2}\right)=P\left(C_{3}\right)=\frac{1}{4} .\) Find \(P\left[\left(C_{1}^{c} \cap C_{2}^{c}\right) \cup C_{3}\right]\)
If \(P\left(C_{1}\right)>0\) and if \(C_{2}, C_{3}, C_{4}, \ldots\) are mutually disjoint sets, show that \(P\left(C_{2} \cup C_{3} \cup \cdots \mid C_{1}\right)=P\left(C_{2} \mid C_{1}\right)+P\left(C_{3} \mid C_{1}\right)+\cdots\)
Let \(X\) be a random variable such that \(E\left[(X-b)^{2}\right]\) exists for all real \(b\). Show that \(E\left[(X-b)^{2}\right]\) is a minimum when \(b=E(X)\).
Let \(X\) be a number selected at random from a set of numbers \(\\{51,52, \ldots, 100\\}\). Approximate \(E(1 / X)\) Hint: Find reasonable upper and lower bounds by finding integrals bounding \(E(1 / X)\).
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