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## Homework Statement

Given:

f(x,y) = x + y, for 0<x<1 and 0<y<1

f(x,y) = 0, otherwise

Derive the joint distribution function of X and Y.

## Homework Equations

N.A.

## The Attempt at a Solution

Using the definition, I obtained part of the joint distribution F(x,y) = (1/2)(xy)(x+y) for 0<x<1, 0<y<1. Leaving out the working for this as it is pretty standard. I am not sure if my next step is correct and so is hoping you guys can help check my logic...

**1)**F(x,y) has to be defined for all (x,y) in R^2. So I have to consider all possible points. Correct?

**2)**When 0<x<1 and y > or = 1, F(x,y) = left limit of F(x,y) as y tend to 1 = F(x,1) = (1/2)(x)(x+1)? I am guessing that values of y above 1 do not affect the joint distribution F(x,y) so it takes the same value as its left limit at the "boundary" of possible y values?

**3)**By symmetry, when x > or = 1 and 0<y<1, F(x,y) = (1/2)(y)(y+1).

**4)**Finally, for x > or = 1 and y > or = 1, F(x,y) = 1. And F(x,y) = 0 for all other values.

Phew. I am most concern about step 2). The rest were included for completeness. Thanks!!!