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 | \(P(X = x) = p_X(x) > 0\)possible | \(P(X = x) = 0\) 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 | \(p_{X,Y}(x,y) = p_X(x) \cdot p_Y(y)\) | \(f_{X,Y}(x,y) = f_X(x) \cdot f_Y(y)\) |