Lesson 1: Continuous Random Variables

Introduction to continuous random variables and PDFs.

Lesson 1: Continuous Random Variables

Probability Density Function (PDF)

For a continuous random variable $X$, the PDF $f(x)$ describes the relative likelihood of the random variable taking a value near $x$. $P(a \le X \le b) = \int_a^b f(x) dx$

Properties:

1. $f(x) \ge 0$
2. $\int_{-\infty}^{\infty} f(x) dx = 1$

Cumulative Distribution Function (CDF)

$F(x) = P(X \le x) = \int_{-\infty}^x f(t) dt$

Expectation and Variance

$E[X] = \int_{-\infty}^{\infty} x f(x) dx$ $Var(X) = E[X^2] - (E[X])^2$