Multivariable Distributions

Discrete

For two discrete random variables $X$ and $Y$ , the joint probability of the events $A = (X = x)$ and $B = (Y = y)$ is

P (X = x, Y = y) = P (X = x and Y = y) = P (A \cap B)

The function of the joint probability is called the joint probability mass function

f (x) = P (X = x, Y = y)

If and only if:

P ((X, Y) \in A) = \sum_{(x, y)} \sum_{\in A} f_{X, Y} (x, y)

If the random variables $X$ and $Y$ are independent, the joint pmf is the product of the two Marginal Distributions

P (X = x, Y = y) = P (X = x) P (Y = y) = f_{X} (x) f_{Y} (y)

If $X$ and $Y$ are discrete random variables and $f (x, y)$ is the joint PMF of $X$ and $Y$

F (x, y) = P (X \leq x, Y \leq y) = \sum_{s \leq x} \sum_{t \leq y} f (s, t) for x, y \in R

A bivariate function $f (x, y)$ of continuous random variable $X$ and $Y$

P [(X, Y) \in A] = \int_{A} \int f_{X, Y} (x, y) d x d y

It can serve as a joint PDF of a pair of continuous random variables $X$ and $Y$ if it satisfies

If $X$ and $Y$ are continuous random variables, the function

F (X, Y) = P (X \leq x, Y \leq y) = \int_{- \infty}^{y} \int_{- \infty}^{x} f (s, t) d s d t x, y \in (- \infty, \infty)

where $f (s, t)$ is the joint pdf of $X$ and $Y$ at $(s, t)$ is called the joint cumulative distribution of $X$ and $Y$

Similar to the single-variable functions,

f (x, y) = \frac{\partial^{2}}{\partial x \partial y} F (x, y)