Thread: Not sure how to set up integral limit on joint density probability

For the given joint probability density f(x,y) = 2 for x>0, y>0, x+y<1; 0 elsewhere: find P(X+Y>0.75).

I got that the limits of the density function itself are {x,0,1} and {y,0,1-x}. For this specifc probability though, x+y<1 AND x+y>0.75, right? Possibly it is {x,0.75,1}, {y,0,1-x} ? I don't think so though, because that assumes x is the greater number here.

Hey drabbie.

For these kinds of problems, it's best to draw the region on a piece of paper and do it this way.

But for your problem x + y < 1 is basically the triangle bounded by the line x + y = 1 and the x and y axes and this gives the limits you got.

However as you have hinted: this region is for the total probability space and you need to find the intersection of the sets where X + Y > 0.75 with X + Y < 1. So in this case you will actually get a "rhombus-like" region since you have to "subtract" the triangle that lies below the line x + y = 0.75 that is also bounded by the x and y axes respectively.

Now instead of going from 0 to 1-x in the y you are going to go from 0.75 - x to 1 - x for the y-region, but your x-region will be unchanged since when x gets to 0.75 you will have X + Y > 0.75 since X > 0.75.

However you have to adjust for when X > 0.75 because if you don't you will be including the area below the x-axis which means that you have two regions: one between 0 and 0.75 and one between 0.75 and 1 for x. The region between 0 to 0.75 is discussed above but the other region is just with the normal limits of 0 to 1-x for x = 0.75 to 1.

So you will have two double integrals with different limits corresponding to x = [0,0.75) and x = [0.75,1]