91Ó°ÊÓ

Suppose that \(r\) of \(N\) chips are red. Divide the chips into three groups of sizes \(n_{1}, n_{2}\), and \(n_{3}\), where \(n_{1}+\) \(n_{2}+n_{3}=N\). Generalize the hypergeometric distribution to find the probability that the first group contains \(r_{1}\) red chips, the second group \(r_{2}\) red chips, and the third group \(r_{3}\) red chips, where \(r_{1}+r_{2}+r_{3}=r\).

An urn contains five balls numbered 1 to 5 . Two balls are drawn simultaneously. (a) Let \(X\) be the larger of the two numbers drawn. Find \(p_{X}(k)\). (b) Let \(V\) be the sum of the two numbers drawn. Find \(p_{V}(k)\).

Suppose a fair die is tossed three times. Let \(X\) be the number of different faces that appear (so \(X=1,2\), or 3 ). Find \(p_{X}(k)\).

A fair coin is tossed three times. Let \(X\) be the number of heads in the tosses minus the number of tails. Find \(p_{X}(k)\).

Suppose that five people, including you and a friend, line up at random. Let the random variable \(X\) denote the number of people standing between you and your friend. What is \(p_{X}(k)\) ?

Suppose \(X\) is a binomial random variable with \(n=4\) and \(p=\frac{2}{3}\). What is the pdf of \(2 X+1\) ?

A fair die is rolled four times. Let the random variable \(X\) denote the number of 6 's that appear. Find and graph the cdf for \(X\).

A fair die is rolled four times. Let the random variable \(X\) denote the number of 6 's that appear. Find and graph the cdf for \(X\).

A random variable \(Y\) has cdf $$ F_{Y}(y)= \begin{cases}0 & y<1 \\ \ln y & 1 \leq y \leq e \\ 1 & e

Recall the game of Keno described in Question 3.2.26. The following are all the payoffs on a \(\$ 1\) wager where the player has bet on ten numbers. Calculate \(E(X)\), where the random variable \(X\) denotes the amount of money won. \begin{tabular}{crc} \hline Number of Correct Guesses & Payoff & Probability \\ \hline\(<5\) & \(-81\) & \(.935\) \\ 5 & 2 & \(.0514\) \\ 6 & 18 & \(.0115\) \\ 7 & 180 & \(.0016\) \\ 8 & 1,300 & \(1.35 \times 10^{-4}\) \\ 9 & 2,600 & \(6.12 \times 10^{-6}\) \\ 10 & 10,000 & \(1.12 \times 10^{-7}\) \\ \hline \end{tabular}