Random variables
| Discrete (Probability Mass Function, p.m.f) | Continuous (Probability Density Function, p.d.f) |
|---|---|
| \(p(x) = \mathbb{P}(X = x)\) | \(f(x)\) |
| \(\forall x \quad 0 \leq p(x) \leq 1\) | \(\forall x \quad f(x) \geq 0\) |
| \(\sum_{\text{all } x} p(x) = 1\) | \(\int_{-\infty}^{\infty} f(x) \, dx = 1\) |
Cumulative distributioin function
\(\text{c.d.f }\)
\(F(x) = \mathbb{P}(X \leq x)\)
| Discrete (c.d.f) | Continuous (c.d.f) |
|---|---|
| \(F(x) = \sum_{y \leq x} p(y)\) | \(F(x) = \int_{-\infty}^{x} f(y) \, dy\) |
Expected value
\(\mathbb{E}(X) = \mathbf{\mu}_X\)
| Discrete Expected value | Continuous Expected value |
|---|---|
| \(\text{If }\underset{\text{all } x} \sum | x | \cdot p(x) < \infty\) | \(\int_{-\infty}^{\infty} | x | \cdot f(x)dx < \infty\) |
| \(\mathbb{E}(X) = \underset{\text{all } x} \sum x \cdot p(x)\) | \(\mathbb{E}(X) = \int_{-\infty}^{\infty} x\cdot f(x)dx\) |
| \(\text{If }\underset{\text{all } x} \sum | g(x) | \cdot p(x) < \infty\) | \(\int_{-\infty}^{\infty} | g(x) | \cdot f(x)dx < \infty\) |
| \(\mathbb{E}(g(X)) = \underset{\text{all } x} \sum g(x) \cdot p(x)\) | \(\mathbb{E}(g(X)) = \int_{-\infty}^{\infty} g(x) \cdot f(x)dx\) |
Variance
\(\text{Var}(X) = \sigma^2_X = \mathbb{E}(\left[ X - \mu_{X} \right]^2) = \mathbb{E}(X^2) - \left[\mathbb{E}(X)\right]^2\)
| Discrete Variance | Continuous Variance |
|---|---|
| \(\text{Var}(X) = \underset{\text{all } x} \sum (x - \mu_X)^2 \cdot p(x)\) | \(\text{Var}(X) = \int_{-\infty}^{\infty} (x - \mu_X)^2 \cdot f(x)dx\) |
| \(= \underset{\text{all } x} \sum x^2 \cdot p(x) - \left[\mathbb{E}(X)\right]^2\) | \(= \left[\int_{-\infty}^{\infty} x^2 \cdot f(x)dx\right] - \left[\mathbb{E}(X)\right]^2\) |
Moment-generating function
\(M_X(t) = \mathbb{E}(\mathcal{e}^{tX})\)
| Discrete moment-generating function | Continuous moment-generating function |
|---|---|
| \(M_X(t) = \underset{\text{all } x} \sum \mathcal{e}^{tx} \cdot p(x)\) | \(M_X(t) = \int_{-\infty}^{\infty} \mathcal{e}^{tx} \cdot f(x)dx\) |
Example 1
| \(x\) | \(p(x)\) | \(F(x)\) |
|---|---|---|
| \(1\) | \(0.2\) | \(0.2\) |
| \(2\) | \(0.4\) | \(0.6\) |
| \(3\) | \(0.3\) | \(0.9\) |
| \(4\) | \(0.1\) | \(1.0\) |
\[ F(x) = \begin{cases} 0 & x < 1 \\ 0.2 & 1 \leq x < 2 \\ 0.6 & 2 \leq x < 3 \\ 0.9 & 3 \leq x < 4 \\ 1 & x \geq 4 \end{cases} \]
library(ggplot2)
# Data
x <- c(1, 2, 3, 4)
Fx <- c(0.2, 0.6, 0.9, 1.0)
# Creating a data frame
data_cdf <- data.frame(x, Fx)
# CDF Plot
ggplot(data_cdf, aes(x = x, y = Fx)) +
geom_step(direction = "hv", linewidth=1) + # Creates the step plot
geom_point(color="red") + # Optional: adds points at steps
ggtitle("Cumulative Distribution Function (CDF)") +
xlab("x") + ylab("F(x)") +
theme_minimal() # Optional: a cleaner theme for the plot