Question

In: Statistics and Probability

Let X and Y have the joint probabilitydensity function (pdf):?(?, ?) = 3/2 ?2(1...

Let X and Y have the joint probability density function (pdf):

f(x, y) = 3/2 x²(1 − y), − 1 < x < 1, − 1 < y < 1

Find P(0 < Y < X).
Find the respective marginal pdfs of X and Y. Are X and Y independent?
Find the conditional pdf of X give Y = y, and E(X|Y = 0.5).

Expert Solution

The joint odf of X,Y can be written as

a) The probability

ans: P(0 < Y < X) = 0.225

b) The marginal pdf of X is

ans: The marginal pdf of X is

The marginal pdf of Y is

ans: The marginal pdf of Y is

Variables, X,Y are independent if the joint pdf of X,Y is equal to the product of marginal pdfs

The product of marginal pdfs of X and Y is

This is equal to the join pdf of X,Y

Hence X,Y are independent

ans: X and Y independent as the joint pdf of X,Y is equal to the product of marginal pdf of X and Y

c) The conditional pdf of X given Y=y is

ans: The conditional pdf of X given Y=y is

Note: This is to be expected, as, since X and Y are independent, the The conditional pdf of X given Y=y is equal to the marginal pdf of X

The expected value is

ans: E(X|Y = 0.5) = 0

orchestra answered 3 years ago

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