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Chapter 2: Problem 22
If a fair coin is successively flipped, find the probability that a head first appears on the fifth trial.
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Suppose that two teams are playing a series of games, each of which is independently won by team \(A\) with probability \(p\) and by team \(B\) with probability \(1-p .\) The winner of the series is the first team to win four games. Find the expected number of games that are played, and evaluate this quantity when \(p=1 / 2\).
Suppose a die is rolled twice. What are the possible values that the following random variables can take on? (a) The maximum value to appear in the two rolls. (b) The minimum value to appear in the two rolls. (c) The sum of the two rolls. (d) The value of the first roll minus the value of the second roll.
Show that the sum of independent identically distributed exponential random variables has a gamma distribution.
If \(X\) is normally distributed with mean 1 and variance 4 , use the tables to
find \(P\\{2
Suppose the distribution function of \(X\) is given by $$ F(b)=\left\\{\begin{array}{ll} 0, & b<0 \\ \frac{1}{2}, & 0 \leq b<1 \\ 1, & 1 \leq b<\infty \end{array}\right. $$ What is the probability mass function of \(X ?\)
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