# 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 repalcement.

• 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 $$0 \le n \le N$$