00:01
In this problem, the joint probability function is given and we have to find the marginal density function of fx y.
00:07
So starting with the first part, we'll find the marginal density function of x.
00:14
So the marginal density function of x will be equal to f x of x and this will be equal to integral from the limit 1 to 2 f xy of xy.
00:42
Multiplied by d, this will be equal to integral from the limit 1 to 2, 3x minus y divided by 9 multiplied by dy.
00:56
And on solving the integral we will get xy divided by 3 minus y squared divided by 18 varying from the limit 1 to 2 and on solving the limits we will get 2x minus 1 divided by 6 and x is varying from 1 to 3.
01:21
So this is the marginal density function of x.
01:25
Next we'll find the marginal density function for y and it will be fy of y that will be equal to integral from the limit 1 to 3 f xy of x.
01:55
X comma y multiplied by d x and it will be equal to integral from limit 1 to 3x minus y divided by 9 multiplied by d x on solving the integral we will get x squared divided by 6 minus x y divided by 9 varying from the limit 1 to 3 and on solving the limits we will get 12 minus 2 y divided by 9 where y varies from 1 to 2.
02:32
So this is the marginal density function for y.
02:37
In the part b we are required to find whether x and y are independent or not.
02:42
So for that we will firstly find the value of f x of x of x multiplied by f y of y and it will be equal to 2 multiplied by 2x minus 1 multiplied by 6 minus y divided by 54 and fx y of xy is given to be 3x minus y divided by 9 as given in the problem we can clearly see that f x of x multiplied by f y of y of y is not equal to f xy of x comma y and this means that x and y are not independent and this is the answer for for the part b of this problem...