Variance of a DRV

Like in the case of empirical data, it is also sensible to ask if there is a corresponding concept of variance in the case of a random variable.

Recall that given empirical data \(x_i\)'s, the variance is defined as

\[\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 = \frac{1}{n} \left[ \sum_{i=1}^{n} (x_i^2 - 2\bar{x}x_i + \bar{x}^2) \right] = \frac{1}{n} \sum_{i=1}^{n} x_i^2 - 2\bar{x} \cdot \frac{1}{n} \sum_{i=1}^{n} x_i + \bar{x}^2 \cdot \frac{1}{n} \sum_{i=1}^{n} 1 = \frac{\sum_{i=1}^{n} x_i^2}{n} - \bar{x}^2.\]

Hinging on this analogy, we propose that the variance of a random variable \(X\) is defined as

\[\sigma^2 = \mathrm{Var}(X) = E(X^2) - [E(X)]^2.\]

Of course, this is equivalent to \(E(X - E(X))^2 = E(X - \mu)^2\).