7.5. 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

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

7.5.1. 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

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

7.5.2. 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

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