Exercises 3.12: 4, 15, 30(a)(b), 31(a), 32

Also:

For each of the following, find the cumulative distribution function for the given random variable:

1. Suppose 4 chips are drawn, without replacement, from an urn with 3 red chips and 2 blue chips. Let $X$ be the number of red chips drawn.
2. $X$ is the number of heads in four tosses of a fair coin.
3. $Y$ is number of aces when two cards are drawn a random from a standard deck of 52 cards.

For Problem Set due 5 October: 30(a)(b) from Exercises 3.12 and 2 from the the problems above

Answers:

1. $F_X(x) = \begin{cases}0,& \text{if } x < 2, \\ \frac{3}{5},& \text{if } 2 \le x < 3, \\ 1,& \text{if } x \ge 3, \\ 0.\end{cases}$
2. $F_X(x) = \begin{cases}0,& \text{if } x < 0, \\ \frac{1}{16},& \text{if } 0 \le x < 1, \\ \frac{5}{16},& \text{if } 1 \le x < 2, \\ \frac{11}{16},& \text{if } 2 \le x < 3, \\ \frac{15}{16},& \text{if } 3 \le x < 4, \\ 1,& \text{if } x \ge 4. \end{cases}$
3. $F_Y(y) = \begin{cases}0,& \text{if } y < 0, \\ \frac{188}{221},& \text{if } 0 \le y < 1, \\ \frac{220}{221},& \text{if } 1 \le y < 2, \\ 1,& \text{if } y \ge 2.\end{cases}$