Joint Probability Density Distribution

The joint probability distribution of two discrete random variables, say X and Y, is the probability that the random variables simultaneously take on certain values, say $x$ and $y$. The probabilities of all possible $(x,y)$ combinations sum to $1$.

The joint probability distribution can be written as the function $Pr(X=x,Y=y)$ , which is the prob that happen together.

Reflect to the table, the probability would simply be the multiply of the 2 independent probability of the event.

For example:

In this table, it is easily to tell that, $Pr(X=0,Y=1)=0.15$ and so on. And next we lead in the Marginal Probability Distribution, it is just the adding up from the Joint Probability Density Distribution.

We denote as:

$$f_{X,Y}(x,y)$$