Joint Distribution

Suppose we have $P(X=x,Y=y)$.

Denote as $f_{X,Y}(x,y)$.

Then we must have

$$\int \int f_{X,Y}(x,y)dxdy=1$$

If the two random variables $X$ and $Y$ are not independent, we cannot write them as the product of $X$ and $Y$ distributions. We must be given the joint pdf $f(x,y)$.

In the case of $X$ and $Y$ being independent, then we can write the joint distribution as a product of their pdfs.

$$\int \int f_X(x)f_Y(y)dxdy$$

Refer to Marginal Distributions