Hint: E[W] = E[g(X,Y)] = Integral ([0,1]x[0,1])*g(X,Y)*f(x,y)dydx.
For this one you need to get the region of integration.
Since both are between 0 and 1 and both are independent, then the limits will be X = 0 to 0.5 and Y = 0.5 to 1.
The rest is just evaluating the integral with those limits.
thank you, but the probability will equal only to the double integral with limits like you mentioned, or it should be equal to the double integral you mentioned and divided by another double integral of P(Y>0.5)=double integral of (x+y) with x from 0 to 1 and y from 0.5 to 1; becasue we have intersection of two probabilites. Can you think about it please and tell me?
What you should do for your own confirmation, is to draw the region that is being integrated on paper and convince yourself from that what the actual limits of integration should be.
If this shows a different answer from mine, then please just point out with some graphical evidence where the inconsistency is.
hi, you are right thnx.
For the same, f(x,y) =x+y, I need to calculate:
1) Pr(X<Y) so my double integral goes from X to 1 for Y, and from 0 to 1 for X, is this right?
2) Pr((X+Y)<1) so my double integral goes from 0 to (Y-1) for X, and from 0 to 1 for Y, is this right?
Thanks in advance