The random variables X and Y have joint probability density...

The random variables X and Y have joint probability density function: f(x, y) = { 3/2(x^2 + y^2) 0 < x < 1, 0 < y < 1, 0 otherwise. (a) Determine whether or not X and Y are independent. (b) It is of interest to know something about the proportion of Z = X + Y, the sum of the two random variables. Find E(X + Y). (c) Find E(XY). (d) Find Var(X), Var(Y), and Cov(X, Y). (e) Find Var(X + Y). (f) Find Var(X - 2Y + 3).

Transcript

00:02 Hello students, given f of x, y is equal to 3 by 2 into x square plus y square 0 less than x less than 1 and 0 less than y less than 1 or 0 otherwise.

00:20 Now part a, marginal density function of x that is f of x is equal to integration y over f of x, y into dy that is equal to integration 0 to 1 3 by 2 into x square plus y square into dy that is equal to 3 by 2 into x square into integration 0 to 1 dy plus integration 0 to 1 y square into dy.

00:55 That is equal to 3 by 2 into x square into y with interval 0 to 1 plus y cube divided by 3 with interval 0 to 1 that is equal to 1 by 2 into 3 square 3 into x square plus 1 where 0 less than x less than 1.

01:15 Now marginal function of y that is f of y is equal to integration over x f of x, y into dx.

01:31 So this is equal to integration 0 to 1 3 by 2 into x square plus y square into dx that is equal to 3 by 2 into integration 0 to 1 x square into dx plus y square into integration to 1 0 to 1 dx which is equal to 3 by 2 into x cube divided by 3 over the interval 0 to 1 plus y square into x over the interval 0 to 1 that is equal to 1 by 2 into 3 y square plus 1 where 0 less than y less than 1.

02:07 Now f of x into f of y is equal to 1 by 2 into 3 x square plus 1 into 1 by 2 into 3 into y square plus 1 that is equal to 1 by 4 into 9 into x square into y square plus 3 x square plus 3 y square plus 1 that is not equal to 3 by 2 x square plus y square.

02:40 Therefore f of x into f of y is not equal to f of x, y therefore x and y are not independent.

02:53 Now part b expectation of x plus y is equal to expectation of x plus expectation of y then expectation of x equal to integration 0 to 1 x into f of x dx that is so this is equal to integration 0 to 1 x divided by 2 into 3 by 3 into x square plus 1 into dx that is equal to 1 by 2 into integration 0 to 1 3 x cube dx plus 1 by 2 into integration 0 to 1 x into dx that is equal to 3 by 8 plus 1 by 4 which is equal to 5 by 8.

03:41 Now expectation of y is equal to integration 0 to 1 y into f of y dy so this is equal to integration 0 to 1 y by 2 into 3 y square plus 1 into dy that is equal to 1 by 2 outside integration 0 to 1 3 y cube into dx plus 1 by 2 integration 0 to 1 y dx that is equal to 3 by 8 plus 1 by 4 that is equal to 5 by 8 therefore expectation of x plus y is equal to 5 by 8 plus 5 by 8 that is equal to 5 by 4...

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