00:01
Hello students, according to the given question they have given the function f of x comma y is equal to c e power minus 2x minus 3y.
00:13
Here 0 less than y less than x are the limits.
00:18
Now we have to find the c value.
00:20
So the total probability minus infinity to infinity minus infinity to infinity f of x comma y d x d x d y is equal to 1 so by applying 0 to infinity integral 0 to x c e power minus 2 x minus 3 y into d x d y is equal to 1 by applying the integral we get integral to 0 to infinity c into e power minus 2 x minus 2x into e power minus 3 y by minus 3 to the limits y is equal to 0 2 x d x is equal to 1.
01:07
Now integral 0 to infinity, 3 by 3 e power minus 2 x into e power minus 3x plus 1 into d x is equal to 1.
01:19
By applying this integral 3 by 3 by 3 into minus e power minus 5 x by minus 5.
01:26
Plus e power minus 2x by minus 2 to the limits 0 to infinity is equal to 1.
01:35
So by applying limits 3 by 0 plus 0 minus 1 by 5 plus 1 which is equal to 1.
01:45
Therefore c will be equal to 10.
01:50
Therefore the function f of x comma y will be equal to 10 into e power minus 2x minus 3 y to the limits 0 less than y less than x.
02:03
In the a bit we have to find probability of x less than 2 comma y less than 1 by 10.
02:11
So integral 0 to 1 by 10 integral y 2 f of x comma y d x dy dy.
02:22
So integral 0 to 1 by 10 integral y 2 10 tending to e power minus 2x minus 3y into dx, dx, dy.
02:34
By applying the integral, integral 0 to 1 by 10, tending to e power minus 3y into e power minus 2x y minus 2 to the limits y to 2 into dy...