91Ó°ÊÓ

A joint probability function.

Hence, the random variable assumes is,

The mean is

$E\left(g\right(X,Y\left)\right)=\underset{z∈g(A,B)}{∑}zP\left(g\right(X,Y)=z)$

Now, observe that every $z∈g(A,B)$has form $z=g(x,y)$for some $xinA$and $yinB$.

Also we have that,

So we finally we have that,

$\underset{z∈g(A,B)}{∑}zP\left(g\right(X,Y)=z)=\underset{x,y}{∑}g(x,y)p(x,y)$

Therefore, we have proved the claimed.

A joint probability mass function is proved as$\underset{z∈g(A,B)}{∑}zP\left(g\right(X,Y)=z)=\underset{x,y}{∑}g(x,y)p(x,y)$.

A joint probability density function

Using the fact that the expectation of random variable can be written as an integral where we integrate , we have that

Also we have that,

which yields that

Changing the order of integration, we have that

so we have proved the claimed.

A joint probability density function is proved as .