00:01
Hi, let's start the solution.
00:02
In this question we have given x and y have joint probability density function and function fx ,y is 8xy, 0 less than x less than y less than 1 and 0 elsewhere.
00:21
Formula for moment generating function m x ,y t1 ,t2 that is equal to e, e power t1 x plus t2 y.
00:35
So that is equal to integration 0 to 1 integration 0 to 1 e power t1 x plus t2 y f x ,y dy dx.
00:49
So that is integration 0 to 1 integration 0 to 1 e power t1 x plus t2 y 8xy dy dx.
01:01
So that is integration 0 to 1 8x e power t1 x integration 0 to 1 y e power t2 y dy dx.
01:13
Now we find integration 0 to 1 y e power t2 y dy using integration by parts.
01:21
So now take u is y and v is e power t2 y.
01:29
So integration 0 to 1 y e power t2 y that into dy that is equal to y integration e power t2 y dy limit 0 to 1 minus integration 0 to 1 e power t2 y d dy of y dy.
01:52
So that is equal to y e power t2 y upon t2 limit 0 to 1 minus integration 0 to 1 1 into e power t2 y into dy.
02:07
So we get that is equal to 1 into e power t2 upon t2 minus 0 minus e power t2 y upon t2 limit 0 to 1 by applying this limit and solving this we get 1 upon t2.
02:26
So m x comma y t1 comma t2 that is equal to 1 upon t2 integration 0 to 1 8x e power t1 x dx.
02:40
So that is 8 upon t2 integration 0 to 1 x e power t1 x dx because we know that integration 0 to 1 x e power t1 x dx and integration 0 to 1 y e power t2 y dy is similar.
03:07
So by using same method integration we get integration 0 to 1 x e power t1 x that is dx equal to 1 upon t1.
03:17
So by using this here we get that is equal to 8 upon t1 into t2.
03:27
So now joint generating joint movement generating function of x comma y m x comma y t1 comma t2 is 8 upon t1 t2.
03:40
Now we find the coefficient of correlation between x and y...