00:01
Hello students, the joint density function of x and y is given as 4x e raised to minus x plus y, 0 less than x, less than y, less than infinity, 0 otherwise.
00:10
In a part of the question, we want to find the marginal density function of x and we want to reduce its mean and variance.
00:17
Marginal density function is given as integral over y f of x y dy which is integral over x to infinity 4x e raised to minus x plus y dy is equal to 4x e raised to minus x integral x to infinity e raised to minus y dy which is calculated as 4x e raised to minus x integral e raised to minus y is minus e raised to minus y from the limit x to infinity.
00:47
On applying the limit, you will get the marginal density function of x as f of x is equal to 4x e raised to minus 2x, 0 less than x, less than infinity.
01:02
Now, we want to reduce its mean and variance.
01:06
Mean can be found from expectation of x.
01:09
Expectation of x is equal to integral over x x into f of x dx which is integral over 0 to infinity 4x square e raised to minus 2x dx.
01:24
Now, we can integrate by using the product rule which is consider 4x square as first function and e raised to minus 2x as second function.
01:35
Then the product rule says that first function into integral of second function minus integral of derivative of first function into integral of second function.
01:44
On applying that, we will get it as minus 4x square e raised to minus 2x by 2 from the limit 0 to infinity plus integral 0 to infinity 4x e raised to minus 2x dx.
02:04
On applying the limit, we will get the first function as 0.
02:08
This 0 plus minus 2x e raised to minus 2x from the limit 0 to infinity.
02:16
Again, we have applied product rule plus 4 by 2 integral 0 to infinity e raised to minus 2x dx.
02:26
Therefore, expectation of x is equal to 4 by 2 into 2 minus e raised to minus 2x from 0 to infinity.
02:36
On applying the limit, we will get the expectation of x is equal to mean as 1...