00:01
Hello, students.
00:02
So, in this question, a joint pdf of f of x comma y is equal to 4xy is given.
00:08
And in this case, x lies between 0 to 1 and also y lies between 0 to 1, right? so we have to find out the value of e of xy, e of x and e of y, right? so let's see how we can solve this.
00:20
To solve the e of xy, at first we will try to solve the value of xy, we have to use the double integral formula.
00:29
Okay, so this is the double integral so the function is xy, f x comma y, and i will integrate it with respect to x and y, right? because this is a double integral case.
00:42
So let's see how we can solve this.
00:44
So i am trying to equilate the formula.
00:47
So this is double integral and inside it there is xy and it will be multiplied by 4xy, right? because the function value of the function f of xy is 4xy.
00:59
So, that's why i have written 4x and dx, zy is there.
01:03
So i am taking out the constant part out of the integral part.
01:07
So 4 is constant and the lower limit should be 0 and upper limit should be 1 in the case of x and y also because the x lies between 0 to 1 and y also lies between 0 to 1.
01:20
This is why i have written this lower limit and upper limit.
01:23
And this will become x squared and this will become y square and it will be integrated with respect to x and y right so next we will solve the integral part so 0 to 1 is the upper limit and lower limit and this will be x square and if we integrate this part this will become y cube divided by 3 and the upper limit is 1 lower limit is 0 and this will be integrated with respect to x right so now we will solve this further so this will become 4 divided by 3 because 1 by 3 is a constant.
01:57
This is why i'm taking it out of the integral part and the lower limit is 0 and upper limit 1 and this is x square and it will be integrated with respect to x, right? so this is 4 divided by 3 and the integral part of this will be x q divided by 3 and upper limit is 1, lower limit is 0.
02:19
So the final value will be 4 divided by 9 right.
02:24
So the final value became 4 divided by 9.
02:27
So, i have to calculate the fx part right now.
02:31
So i'm trying to write the formula of fx.
02:34
So i have to find the formula of fx.
02:38
So this will be the formula of fx, a lower limit should be y and the value of function is 4xy and it is dy, right? we will integrate it with respect to x, this is why i have written as dy, right? so this will become 4x.
02:55
And this will the integration part of y will be y squared divided by 2 and upper limit is 1, lower limit is 0.
03:04
So it will become 2x.
03:05
Why to ask because x lies between 0 to 1, right? so x lies between 0 to 1.
03:12
This is why i have written it as 2x right.
03:16
Now we are going to calculate the value of e of x right.
03:22
So we are trying to write the value of e of x and and this is the formula of e of x.
03:28
So, lower limit would be 0, upper limit will be 1.
03:31
And this the function would be f of x and this will be integrated with respect to x, right? so this was the formula and now i'm trying to solve the integral part.
03:42
So the lower limit is 0, upper limit is 1 and inside is the value of f of x is given in this question.
03:49
So it will become 2x and will integrate it with respect to x, right? so this is why i'm writing as dx.
03:57
So i'm taking out the constant part means 2 is the constant in this case...