Continuous Random Variables

What is a Continuous Random Variable?

A continuous random variable takes values in an interval (uncountable set). Unlike discrete variables, individual points have zero probability—only intervals have positive probability.

Key Difference

• Discrete: $P(X = x)$ can be positive
• Continuous: $P(X = x) = 0$ for any specific value $x$

Probability Density Function (PDF)

The PDF $f_X(x)$ describes how probability is distributed along the real line:

$$P(a \leq X \leq b) = \int_a^b f_X(x) \, dx$$

Properties

1. $f_X(x) \geq 0$ for all $x$
2. $\int_{-\infty}^{\infty} f_X(x) \, dx = 1$ (total probability is 1)

Interpretation

The PDF is not a probability—it’s a density (probability per unit length). The value $f_X(x)$ can be greater than 1!

Small Intervals

For a small interval of length $\delta$:

$$P(x \leq X \leq x + \delta) \approx f_X(x) \cdot \delta$$

Cumulative Distribution Function (CDF)

The CDF gives the probability of being less than or equal to a value:

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

Properties

1. $F_X(x)$ is non-decreasing
2. $\lim_{x \to -\infty} F_X(x) = 0$
3. $\lim_{x \to \infty} F_X(x) = 1$
4. $P(a \leq X \leq b) = F_X(b) - F_X(a)$

Relation to PDF

$$f_X(x) = \frac{d}{dx} F_X(x)$$

The PDF is the derivative of the CDF.

Common Continuous Distributions

Uniform

All values in $[a, b]$ equally likely:

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

Exponential

Time until an event occurs:

$$f_X(x) = \lambda e^{-\lambda x} \quad \text{for } x \geq 0$$

Normal (Gaussian)

Bell-shaped curve:

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

Expected Value and Variance

Expected Value

$$E[X] = \int_{-\infty}^{\infty} x \cdot f_X(x) \, dx$$

Variance

$$\text{Var}(X) = E[(X - E[X])^2] = E[X^2] - (E[X])^2$$

Uniform Example

For $X \sim \text{Uniform}(a, b)$:

• $E[X] = \frac{a + b}{2}$ (midpoint)
• $\text{Var}(X) = \frac{(b-a)^2}{12}$