Introduction to Probability · Problem 4.1.47

Conditional Probability — Grinstead & Snell

Mathematics — Probability · independence-functions-of-random-variables

Problem

Let

X_1

and

X_2

be independent random variables and let

Y_1 = \phi_1(X_1)

and

Y_2 = \phi_2(X_2)

. (a) Show that

P(Y_1 = r, Y_2 = s) = \sum_{\substack{\phi_1(a)=r \\ \phi_2(b)=s}} P(X_1 = a, X_2 = b) .

(b) Using (a), show that

P(Y_1 = r, Y_2 = s) = P(Y_1 = r)P(Y_2 = s)

so that

Y_1

and

Y_2

are independent.