Chapter 1: Problem 10
Two distinct integers are chosen at random and without replacement from the first six positive integers. Compute the expected value of the absolute value of the difference of these two numbers.
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Let \(f(x)=1 / x^{2}, 1
A coin is to be tossed as many times as necessary to turn up one head. Thus the elements \(c\) of the sample space \(\mathcal{C}\) are \(H, T H, T T H\), TTTH, and so forth. Let the probability set function \(P\) assign to these elements the respective probabilities \(\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}\), and so forth. Show that \(P(\mathcal{C})=1 .\) Let \(C_{1}=\\{c:\) c is \(H, T H, T T H\), TTTH, or TTTTH \(\\}\). Compute \(P\left(C_{1}\right)\). Next, suppose that \(C_{2}=\) \(\\{c: c\) is TTTTH or TTTTTH \(\\}\). Compute \(P\left(C_{2}\right), P\left(C_{1} \cap C_{2}\right)\), and \(P\left(C_{1} \cup C_{2}\right)\).
A chemist wishes to detect an impurity in a certain compound that she is making. There is a test that detects an impurity with probability \(0.90\); however, this test indicates that an impurity is there when it is not about \(5 \%\) of the time. The chemist produces compounds with the impurity about \(20 \%\) of the time. A compound is selected at random from the chemist's output. The test indicates that an impurity is present. What is the conditional probability that the compound actually has the impurity?
Find the cdf \(F(x)\) associated with each of the following probability density
functions. Sketch the graphs of \(f(x)\) and \(F(x)\).
(a) \(f(x)=3(1-x)^{2}, 0
The following game is played. The player randomly draws from the set of integers \(\\{1,2, \ldots, 20\\} .\) Let \(x\) denote the number drawn. Next the player draws at random from the set \(\\{x, \ldots, 25\\}\). If on this second draw, he draws a number greater than 21 he wins; otherwise, he loses. (a) Determine the sum that gives the probability that the player wins. (b) Write and run a line of \(\mathrm{R}\) code that computes the probability that the player wins. (c) Write an \(\mathrm{R}\) function that simulates the game and returns whether or not the player wins. (d) Do 10,000 simulations of your program in Part (c). Obtain the estimate and confidence interval, (1.4.7), for the probability that the player wins. Does your interval trap the true probability?
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