Let the random variables X and Y have joint...

Let the random variables $X$ and $Y$ have joint distribution $f(x, y) = egin{cases} frac{3}{8}(x + y^2), & 0 le x le 2, 0 le y le 1\ 0, & ext{otherwise} end{cases}$ Are $X$ and $Y$ independent?

Transcript

00:01 So here we have been given with the function of x comma y, okay, and this is going to be equal to 3 over 8 x plus y square.

00:12 Function of x comma y, this is going to be equal to 3 over 8 x plus y square.

00:20 And this is the case when we have with us, the value of x to be between 0 and 2 and the value of y is between 0 and 1.

00:29 And in other case this value is going to be equal to zero now you need to describe that here x and y are independent okay x and y are going to independent and this case is possible only if we can say that the function of x comma y this function is going to be equal to function of x times function of function of y and if this is the case then only we can say that this function is going to be independent now we are going to calculate our function of x and function of y and then we are going to calculate the function of x times function of y first of all let us calculate our function of x okay the function of x is basically going to be equals to the integration of function of x comma y with respect to d y and the limit is being 0 to 1 okay and we are going to solve this term this term will be 0 to 1 okay the function of x comma y here it is going to be equal to 3 over a x plus y square types dy okay this is the case and here since it is this integration is with respect to y therefore we can say that our variable x is going to be constant and thus our function of x will have the term x and this will be in terms of x and similarly if we are calculating the function of y the function of y will be in respect to y okay and after that what we are going to do is that we are going to multiply the function of x and function of y and again we are going to calculate function of x y and from here we will try to compare whether it is possible to equate them or not therefore it is 3 over 8 okay x plus y is going to of d .y and solving this, solving this term, we will get our result to be 1 over 8, 3x plus 1.

02:37 Okay.

02:38 Now coming to the part of function of y, our function of y is going to be equals to integration from 0 to 2 of function x, y...

Question

Let the random variables $X$ and $Y$ have joint distribution $f(x, y) = egin{cases} frac{3}{8}(x + y^2), & 0 le x le 2, 0 le y le 1\ 0, & ext{otherwise} end{cases}$ Are $X$ and $Y$ independent?

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Let the random variables $X$ and $Y$ have joint distribution $f(x, y) = egin{cases} frac{3}{8}(x + y^2), & 0 le x le 2, 0 le y le 1\ 0, & ext{otherwise} end{cases}$ Are $X$ and $Y$ independent?