05 Joint Distributions

Last Updated: 2 years ago2023-01-25 ; Contributors: AhmedThahir

$$\begin{aligned} f(x, y) &= P(X=x, Y=y) \\ &= P(x \cap y) \\ F(x,y) &= P(X \le x, Y \le y) \\ f(x, y) &\ge 0 \\ f(x|y) &= \frac{f(x, y)}{f(y)} \end{aligned}$$

	Discrete	Continuous
PDF	$\sum\limits_x \sum\limits_y f(x, y) = 1$	$\int\limits_x \int\limits_y f(x, y) \ \mathrm{d} y \mathrm{d} x = 1$
CDF	$\sum\limits_0^x \sum\limits_0^y f(x, y)$	$\int\limits_{- \infty}^x \int\limits_{- \infty}^y f(x, y) \ \mathrm{d} y \mathrm{d} x$
$f(x)$	$\sum\limits_y f(x,y)$	$f(x) = \int\limits_y f(x,y) \ \mathrm{d} y \mathrm{d} x$
$f(y)$	$\sum\limits_x f(x,y)$	$\int\limits_x f(x,y) \ \mathrm{d} y \mathrm{d} x$
$E(x, y)$	$\sum\limits_x \sum\limits_y xy \cdot f(x, y)$	$\int\limits_x \int\limits_y xy \cdot f(x, y) \ \mathrm{d} y \mathrm{d} x$
$E(x)$	$\sum\limits_x x \cdot f(x,y)$	$\int\limits_x x \cdot f(x,y) \ \mathrm{d} y \mathrm{d} x$
$E(y)$	$\sum\limits_y y \cdot f(x,y)$	$\int\limits_y y \cdot f(x,y) \ \mathrm{d} y \mathrm{d} x$

Covariance

$$\begin{aligned} \text{Cov} (x,y) &= E(x,y) - E(x) \cdot E(y) \\ &= \begin{cases} >0 & \text{directly-dependent} \\ 0 & \text{independent}\\<0 & \text{inversely-dependent} \end{cases} \end{aligned}$$

Independence

$$\begin{aligned} f(x,y) &= f(x) \cdot f(y) \\ E(x, y) &= E(x) \cdot E(y) \\ \text{Cov}(x,y) &= 0 \end{aligned}$$

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