Random Variable: Function $X:\Omega \to \mathcal{R}$

Continuous Random Variable

Continuous Random Variable: Random variable with an uncountably infinite range

Cumulative Distribution Function (CDF): $F(x)=P(X\le x)$ for $-\infty < x < \infty$

• $F(-\infty)=\lim_{x\to-\infty} F(x)=0$
• $F(\infty)=\lim_{x \to \infty} F(x) =1$
• $F(x)$ is non-decreasing $\forall x_1 < x_2, F(x_1) \le F(x_2)$

Continuous: Random variable $X$ is continuous if $F(x)$ is continuous for $x\in(-\infty, \infty)$

Probability Density Function (PDF): $f(x)= \frac{dF(x)}{dx}=F'(x)$ is the PDF for $X$

• $\forall x \in (-\infty, \infty), f(x)\ge0$
• $\int_{-\infty}^{\infty}f(x) \, dx = 1$
• $P(X \in [a, b])=\int_a^bf(x) \, dx$

Expected Value: $E(x)=\int_{-\infty}^{\infty}x \cdot f(x) \, dx$

• $E[g(x)]=\int_{-\infty}^{\infty} g(x)f(x) \, dx$

Variance: $V(X) = E(X^2)-E(x)^2$

Uniform Distribution

Uniform PDF: Linear distance between two 1D points

• $uniform(y)= \frac{1}{b-a} \text{ if } (a\le y \le b) \text{ else } 0$
• $E(X)=\frac{a+b}{2}$