00:01
Hello everyone we are going to solve a question and this question we are given a function f of xy is equals to 8 xy where this y is 0 is less than y which is less than x which is less than 1 and this is equals to 0 else way oh so in this question we have to find marginal probability marginal probability density functions, density functions, and also we have to find expected value of x, expected value of y, and the covariance of x and y.
00:51
So this we have to find out.
00:53
Let's start solving the question for the marginal distributions.
00:56
Firstly, we find marginal distribution of x, which is fx.
00:59
So this is integration from 0 to 1, fx, that is 8xy, that is 8xy, d ,y is equal to 8x as it is, the integration of y is y squared divided by 2 and the limits are from 0 to 1.
01:13
So from here we have this is equals to 8 x multiplied with 1 minus 0 and this divided by 2.
01:22
So from here we have this is equals to 4x.
01:26
So this is the answer for the marginal distribution of x.
01:30
Now moving ahead we have to find the marginal distribution of y, which is f of y and this is equals to integration 0 to 1 of 8 x y with respect to d x this is equal to 8 y as it is integration of x is x squared divided by 2 and the limits are from 0 to 1 so we can write this as this is equal to 4 y multiplied with 1 minus 0 which is equal to 4 times y this is the answer for marginal distribution of y now moving ahead we are going to find the expected value of x so the expected value of x that is e of x is equals to integration double integration and the limits are from x is from 0 to 1.
02:13
Y is from 0 to x and here we have x multiplied with function that is 8 x y and the integration is with respect to y and x.
02:22
Now on solving this is equals to 8 times double integration 0 to 1 in 0 to x of x square y, d, d, d x.
02:32
Now firstly we solve the integration with respect to y.
02:36
So this is equal to 8 integration from 0 to 1 x square as it is and here we have y squared divided by 2 limits are from 0 to x here we have d x and on solving this is equal to 4 times integration 0 to 1 x raised to the power 4 d x which is equals to 4 times x raised to the power 5 divided by 5 and the limits are from 0 to 1 and by putting the limit this is equal to 4 divided by 5 which is the answer for expected value of x now we are going to find expected value of y so expected value of y is equals to double integration the y is from 0 to x and x is from 0 to 1 here we have y multiplied with 8 x y and integration is with respect to y and x and now this is equals to on solving this we have this is equals to 0 to 1 integration is from 0 to 1 and this is from 0 to x 8 is constant so this is outside here we have x y square dy d x now integrating the this with respect to y first so this is equal to 8 times 0 to 1 and here we have x multiplied with y cube divided by 3 and the limits are from 0 to x and this is with respect to d x now on solving this is equals to 8 divided by 3 integration from 0 to 1 of x x x .5 .5.
04:06
This is equal to 8 divided by 3 as it is integration is x raised 2 by 5 divided by 5.
04:13
Limits are from 0 to 1.
04:14
And by solving this we have this is equal to 8 divided by 15 and this is the value for expectation value for y.
04:23
Expected value of y...