4.6 Exercises

1. In a box of numbered tickets, 20% of the tickets are numbered 0, 30% are numbered 1, and 50% are numbered 2. Two tickets are drawn from the box at random with replacement. Let $M$ be the maximum and $S$ the sum of the two numbers drawn.

(a) Find the joint distribution of $M$ and $S$.

(b) Find $P(S > 1, M \ge 1)$.

(c) Find the marginal distribution of $S$.

(d) Find the conditional distribution of $S$ given $M=1$.

2. A box contains four tickets labeled $1, 2, 3$, and 4. Suppose you draw tickets one by one at random with replacement.

• Let $X$ be the number of draws until the first time you draw a ticket that you have drawn before.

• Let $Y$ be the number of draws until the second time you draw a ticket that you have drawn before.

For example, if first few draws are $2, 3, 1, 3, 4, 1$ then $X = 4$ and $Y=6$.

(a) What are the possible values $(x, y)$ of the random pair $(X, Y)$?

(b) Find the joint distribution of $X$ and $Y$.

3. A population consists of 10 children, 15 women, and 20 men. I sample 5 people at random without replacement. In the sample, let $K$ be the number of children, $W$ the number of women, and $M$ the number of men.

(a) Find the distribution of $W$.

(b) Find the joint distribution of $K$ and $W$.

(c) Find the conditional distribution of $W$ given that $K=2$.

4. Let $U_1$ and $U_2$ be independent, each uniformly distributed on $1,2,…,n$. Let $S=U_1+U_2$.

(a) Find $P(U_1 = U_2)$.

(b) Use Part (a) and symmetry to find $P(U_1 < U_2)$ and $P(U_1 > U_2)$.

(c) Find the distribution of $S$.

5. Independent random variables $R$ and $S$ have possible values $0, 1, 2, \ldots, N$ for an integer $N > 3$. For $0 \le k \le N$ let $r_k = P(R = k)$ and let $s_k = P(S = k)$.

Write expressions for the following probabilities in terms of $r_0, r_1, \ldots, r_N$ and $s_1, s_2, \ldots, s_N$.

(a) $P(S = R+3)$

(b) $P(S > R+3)$

(c) $P(\max(R, S) \le n)$ for $0 \le n \le N$

(d) $P(\min(R, S) \le n)$ for $0 \le n \le N$