6.4. Two Variables¶

Let $X$ and $Y$ be two random variables and let $F_{(} X, Y) (x, y)$ be their joint CDF.

\begin{array}{r} lim_{\begin{array}{c} x \to - \infty \\ y \to - \infty \end{array}} F_{X, Y} (x, y) = 0. \end{array}

\begin{array}{r} lim_{\begin{array}{c} x \to \infty \\ y \to \infty \end{array}} F_{X, Y} (x, y) = 1. \end{array}

Right continuity:

lim_{x \to x_{0}^{+}} F_{X, Y} (x, y) = F_{X, Y} (x_{0}, y) .

lim_{y \to y_{0}^{+}} F_{X, Y} (x, y) = F_{X, Y} (x, y_{0}) .

The joint probability density function is given by $f_{X, Y} (x, y)$ . It satisfies $f_{X, Y} (x, y) \geq 0$ and

\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} f_{X, Y} (x, y) d y d x = 1.

The joint CDF and joint PDF are related by

F_{X, Y} (x, y) = P (X \leq x, Y \leq y) = \int_{- \infty}^{x} \int_{- \infty}^{y} f_{X, Y} (u, v) d v d u .

Further

P (a \leq X \leq b, c \leq Y \leq d) = \int_{a}^{b} \int_{c}^{d} f_{X, Y} (u, v) d v d u .

The marginal probability is

P (a \leq X \leq b) = P (a \leq X \leq b, - \infty \leq Y \leq \infty) = \int_{a}^{b} \int_{- \infty}^{\infty} f_{X, Y} (u, v) d v d u .

We define the marginal density functions as

f_{X} (x) = \int_{- \infty}^{\infty} f_{X, Y} (x, y) d y

and

f_{Y} (y) = \int_{- \infty}^{\infty} f_{X, Y} (x, y) d x .

We can now write

P (a \leq X \leq b) = \int_{a}^{b} f_{X} (x) d x .

Similarly

P (c \leq Y \leq d) = \int_{c}^{d} f_{Y} (y) d y .

6.4.1. Conditional Density¶

We define

P (a \leq x \leq b | y = c) = \int_{a}^{b} f_{X | Y} (x | y = c) d x .

We have

f_{X | Y} (x | y = c) = \frac{f_{X, Y} (x, c)}{f_{Y} (c)} .

In other words

f_{X | Y} (x | y = c) f_{Y} (c) = f_{X, Y} (x, c) .

In general we write

f_{X | Y} (x | y) f_{Y} (y) = f_{X, Y} (x, y) .

Or even more loosely as

f (x | y) f (y) = f (x, y) .

More identities

f (x | y \leq d) = \frac{\int_{- \infty}^{d} f (x, y) d y}{P (y \leq d)} .

If $X$ and $Y$ are independent then

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

f (x | y) = \frac{f (x, y)}{f (y)} = \frac{f (x) f (y)}{f (y)} = f (x) .

Similarly

f (y | x) = f (y) .

The CDF also is separable

F_{X, Y} (x, y) = F_{X} (x) F_{Y} (y) .