05 Joint Distributions
\[ \begin{aligned} f(x, y) &= P(X=x, Y=y) \\ &= P(x \cap y) \\ F(x,y) &= P(X \le x, Y \le y) \\ f(x, y) &\ge 0 \\ f(x|y) &= \frac{f(x, y)}{f(y)} \end{aligned} \]
| Discrete | Continuous | |
|---|---|---|
| \(\sum\limits_x \sum\limits_y f(x, y) = 1\) | \(\int\limits_x \int\limits_y f(x, y) \ \mathrm{d} y \mathrm{d} x = 1\) | |
| 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¶
\[ \begin{aligned} f(x,y) &= f(x) \cdot f(y) \\ E(x, y) &= E(x) \cdot E(y) \\ \text{Cov}(x,y) &= 0 \end{aligned} \]