A [[Random variable]] with 2 "parts".

Works the same for variables with even higher dimension.

# [[Discrete]] 2D random variable

## Joint probability mass function (pmf)

$$p(x,y) = P(X=x \cap Y=y) \quad \left( = P(X=x, Y=y) \right)$$

## Joint table

Much like the normal joint table but an actual table instead of a single line.

We can get the respective 1D tables by adding rows or columns (depending on
which variable) together.

$$p_X(x) \ \mathrm{of} \ X := p_X(x) = p(x,y_1) + ... + p(x, y_n)$$

## Marginal pdf

# [[Continuous]] 2D random variable

## Joint probability distribution function (pdf)

1D is integrated over the number line, so 2D is integrated over $D$.

$$? = \iint_D f(x,y) dxdy = P((X, Y) \in D)$$

where $D$ is any [[Borel set]] on $\mathbb{R}^2$.

We don't usually draw this graph since it is in 3D (unpleasant).

Instead, we draw the non-trivial domain (all non-zero values).

$$p(x,y) \ge 0, \ \int_{-\infty}^\infty \left( \int_{-\infty}^\infty f(x,y) dx \right) dy = 1$$

## Marginal pdf

$$f_X(x) \ \mathrm{of} \ X := \int_{-\infty}^\infty f(x,y)dy$$

Similar for $f_Y(y)$.

Bounds might be weird.

# Independance

$X$ and $Y$ are independent if

$$p(x,y) = p_X(x) \cdot p_Y(y) \qquad \mathrm{(discrete)}$$
$$f(x,y) = f_X(x) \cdot f_Y(y) \qquad \mathrm{(continuous)}$$

Much the same as independence for [[Event]]s.

Check by multiplying marginal pmf/pdf or if something "looks" dependant. Check
the intuition.