Solution Found!
Let Y1 and Y2 have joint density function f (y1, y2) and
Chapter 5, Problem 43E(choose chapter or problem)
Problem 43E
Let Y1 and Y2 have joint density function f (y1, y2) and marginal densities f1(y1) and f2(y2), respectively. Show that Y1 and Y2 are independent if and only if f (y1|y2) = f1(y1) for all values of y1 and for all y2 such that f2(y2) > 0. A completely analogous argument establishes that Y1 and Y2 are independent if and only if f (y2|y1) = f2(y2) for all values of y2 and for all y1 such that f1(y1) > 0.
Questions & Answers
QUESTION:
Problem 43E
Let Y1 and Y2 have joint density function f (y1, y2) and marginal densities f1(y1) and f2(y2), respectively. Show that Y1 and Y2 are independent if and only if f (y1|y2) = f1(y1) for all values of y1 and for all y2 such that f2(y2) > 0. A completely analogous argument establishes that Y1 and Y2 are independent if and only if f (y2|y1) = f2(y2) for all values of y2 and for all y1 such that f1(y1) > 0.
ANSWER:
Solution:
Step 1 of 2:
Let Y1 and Y2 have joint density function f(y1, y2) and marginal densities f1(y1) and f2(y2), respectively.
We have to show that Y1 and Y2 are independent if and only if f(y2/y1) = f2(y2) for all values of y1 and y2 such that f2(y2)>0.