00:01
Hello students, let's do this question.
00:02
Here given that joint pdf of x and y is given, that is f of xy is equal to x square plus xy upon 3, x ranges from 0 to 1 and y ranges from 0 to 2, and we have to find the given probabilities.
00:14
So first we have to find pdf of x and y respectively.
00:20
So the formula used for finding pdf of x is equal to integration from minus infinity to infinity, f of x, y, d ,y, and pdf of y that is f of y is equal to integration from minus infinity to infinity f of xy d x so first we find the pdfs of x and y therefore f of x is equal to integration from 0 to 2 because we integrate it with respect to y therefore here range of y 0 to 2 x square plus x y upon 3 into dy so after calculating deeds we get the f of x is equal to 2x square plus 2x upon 3 and f of y is equals to integration from 0 to 1 x ranges from 0 to 1 x square plus x y upon 3 d x which equals to 1 by 3 plus y square upon 6 which is pdf of y now probability that x less than or equal to 1 by 2 is equals to integration from 0 to 1 by 2 f of x d x which equals to integration from 0 to 1 by 2 2 x 2 x square plus 2x upon 3 d x so after calculating these we get the value as 2 by 3 into 0 .125 plus 2 by 6 into 0 .0.
02:03
0 .25 which equals to 0 .16666 and probability of y less than 1 by 2 is equals to integration from 0 to 1 by 2 f of y d y equals to integration from 0 to 1 by 2 1 by 3 plus y square upon 6 d y after solving this we get 0 .1 6 plus 0 .00694 is equals to 0 .1736.
02:42
And now we calculate probability that x greater than 1 by 2, which equals to 1 minus probability that x less than or equal to 1 by 2 is equals to 1 minus 0 .1666 which equals to 0 .8333.
02:59
Then probability that y less than x is equals to integration from minus infinity to infinity integration from minus infinity to x f of x y, d y, d x, which equals to integration from 0 to 1 integration from 0 to 2 x square plus x y upon 3 into dy d x which equals to first we integrate with respect to y and then then with respect to x.
03:32
Integrate with respect to y, then we get 0 to 1 x square y plus x by 6 y square 0 to 2 d x and after solving these we get integration from 0 to 1 x square into 2 plus 4x by 6 d x so after calculating these we get 8 by 6...