Chapter 1: Problem 21
If the pdf of \(X\) is \(f(x)=2 x e^{-x^{2}}, 0
Chapter 1: Problem 21
If the pdf of \(X\) is \(f(x)=2 x e^{-x^{2}}, 0
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Get started for freeA hand of 13 cards is to be dealt at random and without replacement from an ordinary deck of playing cards. Find the conditional probability that there are at least three kings in the hand given that the hand contains at least two kings.
Let \(X\) be a positive random variable; i.e., \(P(X \leq 0)=0\). Argue that (a) \(E(1 / X) \geq 1 / E(X)\) (b) \(E[-\log X] \geq-\log [E(X)]\) (c) \(E[\log (1 / X)] \geq \log [1 / E(X)]\) (d) \(E\left[X^{3}\right] \geq[E(X)]^{3}\).
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 \((X+2)^{2}=X^{2}+4 X+4\)
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.
A bowl contains ten chips numbered \(1,2, \ldots, 10\), respectively. Five chips are drawn at random, one at a time, and without replacement. What is the probability that two even-numbered chips are drawn and they occur on even- numbered draws?
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