$p_{X,Y}(x,y) = P(X = x,\ Y = y)$ $\text{where } F_{X,Y}(x,y) = \sum_{s \leq x}\sum_{t \leq y} p_{X,Y}(s,t)$ The Joint Probability Function (JPF) of two discrete random variables $X$ and $Y$ assigns a probability to each pair $(x, y)$, satisfying $p_{X,Y}(x,y) \geq 0$ and $\sum_x \sum_y p_{X,Y}(x,y) = 1$. The Joint Cumulative Distribution Function (JCDF) $F_{X,Y}(x,y)$ gives the probability that $X \leq x$ and $Y \leq y$ simultaneously. > [!example]- Joint Distribution of Two Dice {💡 Example} > Let $X$ be the result of die 1 and $Y$ the result of die 2, each uniform on $\{1,\ldots,6\}$. Find $p_{X,Y}(2,3)$ and $F_{X,Y}(2,3)$. > > > [!answer]- Answer > > Since the dice are independent: > > $p_{X,Y}(2,3) = P(X=2)\cdot P(Y=3) = \frac{1}{6}\cdot\frac{1}{6} = \frac{1}{36}$ > > For the JCDF: > > $F_{X,Y}(2,3) = P(X \leq 2,\ Y \leq 3) = \frac{2}{6}\cdot\frac{3}{6} = \frac{1}{6}$