00:02
In this question, solution here is for the a part we have to find here probability of x greater than y.
00:10
So probability of x greater than y is equals to integration of limit 0 to infinity again integration of limit y to infinity 6 6 multiply by e to the power minus of 2x plus 3y dx dy which is equals to integration of limit 0 to infinity 3 multiply by e to the power minus 3 multiply by minus of e to the power minus 2x x equals to y up to x equals to infinity dy.
00:52
So this is equals to integration of limit 0 to infinity 3 multiply by e to the power minus 3y multiply by e to the power minus 2y dy on solving we will get this equals to integration of limit 0 to infinity 3 multiply by e to the power minus 5y dy on solving we will get here probability of x greater than y is equals to 3 divided by 5.
01:20
Now here probability of x plus y less than equals to 1 is equals to integration of limit 0 to 1 again integration of limit 0 to 1 minus x 6 multiply by e to the power minus of 2x plus 3y dy dx.
01:43
So this is equals to integration of limit 0 to 1 2 multiply by e to the power minus 2x multiply by minus of e to the power minus 3y y is equals to 0 up to y is equals to 1 minus x dx which is equals to integration of limit 0 to 1 2 multiply by e to the power minus 2x multiply by 1 minus e to the power minus 3 multiply by 1 minus x dx.
02:14
So here we get probability of x plus y less than equals to 1 is equals to 1 plus 2 e to the power minus 3 minus 3 e to the power minus 2.
02:29
So this is equals to 1 plus 0 .09957 minus 0 .40601.
02:41
So this is equals to 0 .69356.
02:46
Therefore, probability of x plus y less than equals to 1 is equals to 0 .69356.
02:55
This is the answer for first part.
02:58
Now for the second part we have to find here probability of minimum x comma y greater than equals to 1.
03:09
So the event minimum x comma y greater than equals to 1 is the same as the event x greater than equals to 1 comma y greater than equals to 1.
03:20
Thus probability of minimum x comma y greater than equals to 1 is equals to integration of limit 1 up to infinity again integration of limit 1 up to infinity 6 multiply by e to the power minus 2x plus 3y dy dx...