Chapter 1: Problem 5
Let us select five cards at random and without replacement from an ordinary deck of playing cards. (a) Find the pmf of \(X\), the number of hearts in the five cards. (b) Determine \(P(X \leq 1)\).
Chapter 1: Problem 5
Let us select five cards at random and without replacement from an ordinary deck of playing cards. (a) Find the pmf of \(X\), the number of hearts in the five cards. (b) Determine \(P(X \leq 1)\).
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Get started for freeLet \(X\) have the pdf \(f(x)=2 x, 0
Let \(X\) be a random variable of either type. If \(g(X) \equiv k\), where \(k\) is a constant, show that \(E(g(X))=k\).
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 \(P(X \leq 4)\).
Let the random variable \(X\) have mean \(\mu\), standard deviation \(\sigma\), and
mgf \(M(t),-h
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