Let Y1 and Y2 have joint density function f (y1, y2) and

Chapter 5, Problem 43E

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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.

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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.


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