Random Variables
Learning Objectives
Learning objectives of the Random Variables section.
Summary Table
Summary of the Random Variables section.
| Concept | Discrete Random Variables | Continuous Random Variables |
|---|---|---|
| Sample Space | Countable outcomes | Uncountable outcomes |
| Domain of Variable | x \in \mathbb{Z} | x \in \mathbb{R} |
| Mapping | X: \text{Outcome} \rightarrow x \in \mathbb{Z} | X: \text{Event} \rightarrow x \in \mathbb{R} |
| Probability Function | Probability Mass Function (PMF):p_X(x) = P(X = x) | Probability Density Function (PDF):f_X(x) \geq 0 |
| Total Probability | \sum_x p_X(x) = 1 | \int_{-\infty}^{\infty} f_X(x) \, dx = 1 |
| CDF (Cumulative Distribution) | F_X(x) = \sum_{k \leq x} p_X(k) | F_X(x) = \int_{-\infty}^{x} f_X(u) \, du |
| Probability of Exact Value | P(X = x) = p_X(x) > 0possible | P(X = x) = 0 for any exact x |
| Expectation | \mathbb{E}[X] = \sum_x x \, p_X(x) | \mathbb{E}[X] = \int_{-\infty}^{\infty} x \, f_X(x) \, dx |
| Variance | \text{Var}[X] = \sum_x (x - \mathbb{E}[X])^2 \, p_X(x) | \text{Var}[X] = \int_{-\infty}^{\infty} (x - \mathbb{E}[X])^2 \, f_X(x) \, dx |
| Joint Distribution | Joint PMF:p_{X,Y}(x, y) = P(X = x, Y = y) | Joint PDF:f_{X,Y}(x, y) |
| Marginal Distribution | p_X(x) = \sum_y p_{X,Y}(x, y) | f_X(x) = \int_{-\infty}^{\infty} f_{X,Y}(x, y) \, dy |
| Conditional Probability | p_{X|Y}(x|y) = \frac{p_{X,Y}(x,y)}{p_Y(y)} | f_{X|Y}(x|y) = \frac{f_{X,Y}(x,y)}{f_Y(y)} |
| Conditional Expectation | \mathbb{E}[X|Y=y] = \sum_x x \, p_{X|Y}(x|y) | \mathbb{E}[X|Y=y] = \int x \, f_{X|Y}(x|y) \, dx |
| Independence | p_{X,Y}(x,y) = p_X(x) \cdot p_Y(y) | f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y) |