Chapter 1: Problem 3
Let \(X\) have the pdf \(f(x)=(x+2) / 18,-2
Chapter 1: Problem 3
Let \(X\) have the pdf \(f(x)=(x+2) / 18,-2
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Get started for freeLet \(X\) 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.
Find the 25 th percentile of the distribution having pdf \(f(x)=|x| / 4,-2<\) \(x<2\), zero elsewhere.
A pair of dice is cast until either the sum of seven or eight appears. (a) Show that the probability of a seven before an eight is \(6 / 11\). (b) Next, this pair of dice is cast until a seven appears twice or until each of a six and eight have appeared at least once. Show that the probability of the six and eight occurring before two sevens is \(0.546\).
A die is cast independently until the first 6 appears. If the casting stops on an odd number of times, Bob wins; otherwise, Joe wins. (a) Assuming the die is fair, what is the probability that Bob wins? (b) Let \(p\) denote the probability of a \(6 .\) Show that the game favors Bob, for all \(p\), \(0
A coin is tossed two independent times, each resulting in a tail (T) or a head (H). The sample space consists of four ordered pairs: TT, TH, HT, HH. Making certain assumptions, compute the probability of each of these ordered pairs. What is the probability of at least one head?
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