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