20.4 Variance

Plainly speaking, the expected value measures the average value of the random variable. However, even though both Uniform(−1,1) and Uniform(−100,100) have zero expected value, the latter is much more spread out than the former. Thus, 𝔼[X] is not a good descriptor of the random variable X.

To add one more layer, we measure the average deviation from the expected value. This is done via the variance and the standard deviation.

Definition 94. (Variance and standard deviation)

Let (Ω,Σ,P) be a probability space, let X : Ω →ℝ be a random variable, and let μ = 𝔼[X] be its expected value. The variance of X is defined by

while its standard deviation is defined by

Take note that in the literature, the expected value is often denoted by μ, while the standard deviation is denoted by σ. Together, they form two of the most important descriptors of a random variable.

Figure 20.2 shows a visual...