6.2 Exercise 7 — Applied Probability

Suppose $X , Y$ in canonical form are

X = 2 I_{A_{1}} + 3 I_{A_{2}} + 5 I_{A_{3}} Y = I_{B_{1}} + 2 I_{B_{2}} + 3 I_{B_{3}}

The $P \left(A_{i}\right)$ are 0.3, 0.6, 0.1, respectively, and the $P \left(B_{j}\right)$ are 0.2 0.6 0.2. Each pair $\left\{A_{i} , B_{j}\right\}$ is independent. Consider the random variable $Z = X + Y$ . Then $Z = 2 + 1$ on $A_{1} B_{1}$ , $Z = 3 + 3$ on $A_{2} B_{3}$ , etc. Determine the value of Z on each $A_{i} B_{j}$ and determine the corresponding $P \left(A_{i} B_{j}\right)$ . From this, determine the distribution for Z.

Pfeiffer, Applied Probability, 6.2 Problems on Random Variables and Probabilities, exercise 7

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