Two variables

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

\[\begin{split}\lim_{\substack{x \to -\infty\\ y \to -\infty}} F_{X, Y} (x, y) = 0.\end{split}\]

\[\begin{split}\lim_{\substack{x \to \infty\\ y \to \infty}} F_{X, Y} (x, y) = 1.\end{split}\]

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

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

Further

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

The marginal probability is

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

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

\[\PP (a \leq X \leq b) = \int_{a}^{b} f_X(x) d x.\]

Similarly

\[\PP (c \leq Y \leq d) = \int_{c}^{d} f_Y(y) d y.\]

Conditional density

We define

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

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

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

Independent variables

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

\[F_{X, Y}(x, y) = F_X(x) F_Y(y).\]