PROBLEM I (25 points) The joint probability density function...

PROBLEM I (25 points) The joint probability density function (PDF) of two random variables (X, Y) is given by f_XY(x, y) = { 1, 0 <= x <= 2, 0 <= y <= 1, 2y <= x 0, otherwise } 1) Find the correlation coefficient rho_XY between the two random variables X and Y. 2) Find the probability P(Y>X/2).

Transcript

00:01 In this question, the joint property density function of x, y of x, comma, y is given as 1 when 0 less than or equal to x less than or equal to 2 and 0 less than or equal to 1 comma 2 y less than or equal to x and 0 otherwise.

00:22 This is the joint property density function.

00:24 First we want to calculate the correlation coefficient.

00:28 The first question is we want to calculate the correlation.

00:31 Coefficient row xy so for this we need to find covariance of x comma y and variance of x and variance of y so what is covariance of x comma y which is equal to e of x y minus e of x multiplied by e of y and what is variance of x e of x square minus e of x the whole square.

01:01 And variance of y is e of y square minus e of y the whole square.

01:06 Okay.

01:08 Now what we have to find e of x, e of y, e of x square, e of y square and e of x y of x.

01:18 All these things we need to find it out now.

01:20 So from this.

01:21 First we will find the marginal density function of x.

01:24 Y so f x of x equal to integral 0 to 1 b y because the function f of x comma y is given as 1 okay so which is y from 0 to 1 which is equal to 1 next one f y of y equal to y equal to integral 0 to 2 1 into b x so x from 0 to 2 so which implies 2 right next e of x which is equal to integral 0 to 2 x into x into fx of x 1 into d x so which is equal to x square by 2 from 0 to 2 so which is equal to 4 by 2 which is 2 then e of x square which is equal to integral 0 to 2 x squared by d x multiplied by d x multiplied by d x so which is equal to x cube by 3 between the limit 0 and 2 so which is 8 by 3 next e of y which is equal to integral 0 to 1 y into f y of y is 2 2 into d y so which is equal to 2 y square by 2 from 0 to 1 so here 2 we can cancel and by substituting the limits we will get the value for this as 1 next e of y square which is equal to integral 0 to 1 y square multiplied by 2 d y which is equal to 2 y square by 3 from 0 to 0 2 2 2 2 2 2 2 2 2 2 2 2 2 square by 3 from 0 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 which is equal to 2 by 3.

02:57 Meant e of xy which is equal to integral 0 to 1 integral 0 to 2 xy d x dy.

03:07 So which is equal to first we will integrate with respect to x so y we will take outside so x d x is x squared by 2 from 0 to 2 d y which is equal to integral 0 to 1 y multiplied by while substituting the limit we will get this as 2 .2 .2.

03:28 So which is equal to again integrate this 2 y square by 2 from 0 to 1...

Question

PROBLEM I (25 points) The joint probability density function (PDF) of two random variables (X, Y) is given by f_XY(x, y) = { 1, 0 <= x <= 2, 0 <= y <= 1, 2y <= x 0, otherwise } 1) Find the correlation coefficient rho_XY between the two random variables X and Y. 2) Find the probability P(Y>X/2).

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