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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 \(0 \le n \le N\)