Chapter 1: Problem 12
Let \(X\) be a random variable such that \(R(t)=E\left(e^{t(X-b)}\right)\) exists
for \(t\) such that \(-h
Chapter 1: Problem 12
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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Get started for freeLet \(X\) have the pmf \(p(x)=1 / k, x=1,2,3, \ldots, k\), zero elsewhere. Show that the mgf is $$M(t)=\left\\{\begin{array}{ll}\frac{e^{t}\left(1-e^{k t}\right)}{k\left(1-e^{t}\right)} & t \neq 0 \\ 1 & t=0\end{array}\right.$$
Let \(X\) be a random variable such that \(P(X \leq 0)=0\) and let \(\mu=E(X)\) exist. Show that \(P(X \geq 2 \mu) \leq \frac{1}{2}\).
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 \(P\left[\left(C_{1}^{c} \cap C_{2}^{c}\right) \cup C_{3}\right]\)
Let \(X\) have the pdf \(f(x)=3 x^{2}, 0
Divide a line segment into two parts by selecting a point at random. Find the probability that the larger segment is at least three times the shorter. Assume a uniform distribution.
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