Fundamentals of Applied Probability and Random Processes, 2nd Edition

5.11 Problems

Section 5.3 Discrete Bivariate Random Variables

5.1 The joint PMF of two discrete random variables X and Y is given by

${\mathit{p}}_{\mathit{XY}}\left(\mathit{x},\mathit{y}\right)=\left\{\begin{array}{l}\mathit{kxy}\mathit{x}=1,2,3;\mathit{y}=1,2,3\\ 0\mathrm{otherwise}\end{array}\right.$

a. Determine the value of the constant k.

b. Find the marginal PMFs of X and Y.

c. Find P[1 ≤ X ≤ 2, Y ≤ 2].

5.2 A fair coin is tossed three times. Let the random variable X denote the number of heads in the first two tosses, and let the random variable Y denote the number of heads in the third toss. Determine the joint PMF p_XY(x, y) of X and Y.

5.3 The joint PMF of two random variables X and Y is given by

${\mathit{p}}_{\mathit{XY}}\left(\mathit{x},\mathit{y}\right)=\left\{\begin{array}{l}0.10\mathit{x}=1,\mathit{y}=1\\ 0.35\mathit{x}=2,\mathit{y}=2\\ 0.05\mathit{x}=3,\mathit{y}=3\\ 0.50\mathit{x}=1,\mathit{y}=1\\ 0\mathrm{otherwise}\end{array}\right.$

a.