Solution Found!
Prove that the results in Exercise 5.43 also hold for
Chapter 5, Problem 44E(choose chapter or problem)
Problem 44E
Prove that the results in Exercise 5.43 also hold for discrete random variables.
Reference
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 44E
Prove that the results in Exercise 5.43 also hold for discrete random variables.
Reference
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 44E
Step1 of 2:
Let us consider a random variables () have joint density function
and marginal densities
.
We need to show that this also hold for discrete random variables. That is and
are independent if and only if
|
=
for all values of
and for all
such that