Continuous Random Variables

Definition

A continuous random variable can take any value in an interval. Examples: height, weight, temperature, time.

Probability Density Function (PDF)

For a continuous random variable, probability is represented by area under a curve called the probability density function $f(x)$.

Properties of a PDF:

1. $f(x) \geq 0$ for all $x$
2. Total area under the curve equals 1: $\int_{-\infty}^{\infty} f(x) \, dx = 1$
3. $P(a \leq X \leq b) = \int_a^b f(x) \, dx$ (area under curve from $a$ to $b$)

Key Property

For continuous random variables: $P(X = a) = 0 \text{ for any single value } a$

Therefore: $P(X \leq a) = P(X < a)$

This is different from discrete random variables!

Uniform Distribution

The simplest continuous distribution. $X \sim \text{Uniform}(a, b)$:

$f(x) = \frac{1}{b-a} \text{ for } a \leq x \leq b$

$\mu = \frac{a + b}{2}, \quad \sigma = \frac{b - a}{\sqrt{12}}$

$P(c \leq X \leq d) = \frac{d - c}{b - a}$

Cumulative Distribution Function (CDF)

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

Properties:

• $F(x)$ is non-decreasing
• $0 \leq F(x) \leq 1$
• $P(a < X < b) = F(b) - F(a)$

Mean and Variance

$\mu_X = E(X) = \int_{-\infty}^{\infty} x \cdot f(x) \, dx$

$\sigma_X^2 = \int_{-\infty}^{\infty} (x - \mu)^2 \cdot f(x) \, dx$

Normal Distribution Revisited

The Normal distribution $N(\mu, \sigma)$ is the most important continuous distribution:

$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$

AP Tip: For continuous distributions, always think "area = probability." For the AP exam, you mainly need Normal and Uniform distributions.