00:01
Hello students, in this question, we have given the joint density function of the two random variables x and y and by using that in the first question, we have to calculate the conditional distribution of the x given y.
00:14
Now for that first we need to calculate the marginal pdf of x and y.
00:20
So f of x which is the marginal pdf of x is equal to integration over y from 0 to infinity f of xy dy and now this is equal to here.
00:36
We are taking the integration with respect to y.
00:39
So that means 7 into x is constant so we can take it outside the integration and then integration from 0 to infinity e raised to minus x into y plus 7 dy and now this is equal to 7 into x now integration of e raised to minus x into y plus 7.
01:00
Is e raised to minus x into y plus 7 divide by minus x and limits from 0 to infinity.
01:10
Then after putting the limits we will get this is equal to 7 into e raised to minus 7 x.
01:18
This is the marginal density function of x similarly f of y is equal to 7 into integration from 0 to infinity x into e raised to minus x into y plus 7 into dx and now this is equal to 7 into now this we can write it as gamma 2 divide by y plus 7 raised to 2.
01:47
This is because we know the gamma function that is integration from 0 to infinity x raised to n minus 1 into e raised to minus ax dx is equal to gamma n divide by a raised to n.
02:04
So by using this formula, we will get this and after simplifying this we will get this is equal to 7 divide by y plus 7 raised to and now by using this we can calculate f of x given y is so formula for the conditional density function is f of xy divide by f of y...