Two fundamental concepts in probability theory are expectation (mean) and variance.

• The expectation represents the “average value” of a random variable.
• The variance measures how much the values spread out around the expectation.

In this article, we explain how to compute them in both discrete and continuous settings, with simple examples.

Definition of Expectation

Discrete Case

If a random variable $X$ takes values $x_i$ with probabilities $p_i$:

This is a weighted average.

Continuous Case

If $X$ has probability density function $f(x)$:

This is the average with respect to the density.

Definition of Variance

Variance is defined as the expectation of the squared deviation from the mean:

It can be rewritten in a more convenient form:

• $E[X^2]$: the expectation of the square
• $(E[X])^2$: the square of the expectation

Example 1: A Die (Discrete Uniform Distribution)

Let $X$ be the outcome of a fair six-sided die ($1,2,3,4,5,6$).

• Expectation:

• Variance:

Example 2: Uniform Distribution on [0,1] (Continuous)

Let $X$ be uniformly distributed on [0,1].
Its density is $f(x) = 1$ for $0 \leq x \leq 1$.

• Expectation:

• Variance:

Summary

• Expectation is the average value of a random variable.
  • Discrete: $E[X] = \sum x_i p_i$
  • Continuous: $E[X] = \int x f(x) dx$
• Variance measures the spread around the expectation.
  • $V[X] = E[X^2] - (E[X])^2$
• Example: Die → $E[X]=3.5, \; V[X]\approx 2.92$
• Example: Uniform[0,1] → $E[X]=0.5, \; V[X]\approx 0.083$

Understanding expectation and variance deepens our grasp of how probability distributions are characterized. These quantities will keep appearing as we study specific distributions in more detail.