Expected Value and Variance for Continuous RVs

Expected Value

For a continuous random variable $X$ with PDF $f_X(x)$:

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

Intuition

• Still the “center of mass” or “center of gravity” of the PDF
• Represents the long-run average if you repeat the experiment many times
• Same interpretation as for discrete random variables

Expected Value of Functions

For a function $g(X)$:

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

No need to find the PDF of $g(X)$ first!

Key Insight

Same as discrete case: $E[g(X)] \neq g(E[X])$ in general. Cannot interchange expectation and non-linear functions.

Variance

$$\text{Var}(X) = E\left[(X - E[X])^2\right] = \int_{-\infty}^{\infty} (x - \mu)^2 f_X(x) \, dx$$

Alternative Formula

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

This is often easier to compute.

Standard Deviation

$$\sigma_X = \sqrt{\text{Var}(X)}$$

Uniform Distribution Example

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

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

Expected Value

By symmetry (or integration):

$$E[X] = \frac{a + b}{2}$$

The mean is the midpoint of the interval.

Variance

$$\text{Var}(X) = \frac{(b-a)^2}{12}$$

Standard Deviation

$$\sigma_X = \frac{b-a}{\sqrt{12}} \approx \frac{b-a}{3.46}$$

The standard deviation is proportional to the width of the interval.

Properties (Same as Discrete)

Linearity

• $E[aX + b] = aE[X] + b$
• $E[X + Y] = E[X] + E[Y]$ (always)

Variance Properties

• $\text{Var}(aX) = a^2 \text{Var}(X)$
• $\text{Var}(X + b) = \text{Var}(X)$
• $\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y)$ if $X, Y$ independent

Cumulative Distribution Function (CDF)

The CDF unifies discrete and continuous cases:

$$F_X(x) = P(X \leq x)$$

For Continuous RVs

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

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.

Conditional Expectation

Given an event $A$:

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

Same concept as discrete case: expected value in the “conditional universe.”