Thread: How to set limits of integration

To find P{X + Y > 3}, where X & Y are jointly distributed random variables, I set the limits of integration to 0-to-1 for dx and 3-x to 5 for dy. This seems reasonable given

0 < x < 1
1 < y < 5

Now, for another question, where I find P{X + Y < 1}, I am told the limits of integration should be from 0 to 1-x for dy and 0-to-1 for dx.

I don' see why,
wirefree

Originally Posted by wirefree

To find P{X + Y > 3}, where X & Y are jointly distributed random variables, I set the limits of integration to 0-to-1 for dx and 3-x to 5 for dy. This seems reasonable given

0 < x < 1
1 < y < 5

Mr F says: The above correct.

Now, for another question, where I find P{X + Y < 1}, I am told the limits of integration should be from 0 to 1-x for dy and 0-to-1 for dx.

Mr F asks: For what values of x and y is the joint pdf of X and Y non-zero? If they're the same as above then Pr(X + Y < 1) is obviously zero ....

I don' see why,
wirefree

..

CORRECTION:

For P{X + Y < 1}, I am told the limits of integration should be from 0 to 1-x for dy and 0-to-1 for dx given

0 < x < 1
0 < y < 2

Remain in doubt,
wirefree

Originally Posted by wirefree

CORRECTION:

For P{X + Y < 1}, I am told the limits of integration should be from 0 to 1-x for dy and 0-to-1 for dx given

0 < x < 1
0 < y < 2

Remain in doubt,
wirefree

You're integrating the joint pdf of X and Y over the triangular region bounded by the x- and y-axes and the line y = 1 - x.

NB: X + Y < 1 => Y < 1 - X. The solution to this inequality is the region of the XY-plane that lies below the line .....