00:01
Hello students, in this question we have to find out the k value.
00:03
So here the f of x, y equal to k into 6x plus dy square into dx into dy.
00:10
So it is equal to 1.
00:12
We know that f of x, y into dx into dy equal to 1.
00:24
So we are equating this joint pdf to 1.
00:28
And after that we get k equal to, after simplifying this we get k equal to 5 by 18.
00:33
And after that we have to find out the marginal density f1 of x.
00:38
And to find the marginal density of x we need to integrate the joint density over all possible values of y.
00:48
So here f1 of x equal to integral f of x, y into dy from 0 to 1.
00:59
So after simplifying this we get f1 of x equal to 5 by 18 into 6x plus 1 for 0 less than x less than 2.
01:18
So here for 0 less than x less than 2 or equal to 0 otherwise.
01:36
Or equal to 0 otherwise.
01:38
And now we have to find out the f1 of f2 of x.
01:42
So here the marginal density of, marginal density of f2 y, f2 y.
01:51
So it is equal to integral f of x, y into dx from 0 to, from 0 to 2.
02:00
And here f2 of y equal to 5 by 18 into 12 plus 3y square.
02:07
So for 0 less than y less than 1.
02:11
So equal to 0 otherwise.
02:15
And now we have to determine x and y are independent.
02:19
To determine if x and y are independent we need to check if the joint density can be expressed as the product of the marginal density.
02:26
So here f of x, y equal to f1 of x into f2 of y.
02:33
So let's check f1 of x into f2 of y.
02:37
It is equal to 5 by 18 into 6x plus 1 into 12 plus 3y square divided by 18.
02:48
So this is not equal to the joint density function.
02:51
Therefore x and y are not independent.
02:55
Are not independent...