Variance

Definition

Given an integrable random variable $X$ we define the variance of $X$ as follows:

• $$\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]$$ where $\mu$ is the mean or expected value of $X$

Other forms

Theorem:

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

---

• $$\text{Var}(X)=\mathbb{E}\left[(X-\mu)^2\right]$$

$$=\mathbb{E}\left[X^2-2X\mu+\mu^2\right]$$

$$=\mathbb{E}\left[X^2\right]-2\mu\mathbb{E}[X]+\mu^2$$

But! $$\mu=\mathbb{E}[X]$$

$$=\mathbb{E}\left[X^2\right]-2\mu^2+\mu^2$$

$$=\mathbb{E}\left[X^2\right]-\mu^2$$

$$=\mathbb{E}\left[X^2\right]-(\mathbb{E}[X])^2$$

As required.