# Expectation and variance of a discrete RV

> X takes values {0, 1, 2} with probabilities {0.2, 0.5, 0.3}. Compute E[X] and Var(X) = E[X²] − (E[X])².

Canonical: https://learn.significanthobbies.com/curriculum/drills/compute-expectation-variance

X takes values {0, 1, 2} with probabilities {0.2, 0.5, 0.3}. Compute E[X] and Var(X) = E[X²] − (E[X])².

## The mechanism

A random variable is a function from outcomes to numbers. Expectation is the long-run average; variance measures spread. The CLT says sums of many independent pieces look Gaussian — which is why normal distributions are everywhere.

## Expected evidence

E[X] = 1.1, E[X²] = 1.5, Var(X) = 0.29.

## Hints

- E[X] = Σ x·P(X=x).
- Compute E[X²] before subtracting the squared mean.

## Sources and practice

- [MIT 6.041SC — Probabilistic Systems Analysis (course)](https://ocw.mit.edu/courses/6-041sc-probabilistic-systems-analysis-and-applied-probability-fall-2013/)
- [StatQuest — Probability vs Likelihood](https://www.youtube.com/watch?v=pYx9wEc34tY)
- [Study Random Variables & Distributions](https://learn.significanthobbies.com/curriculum/concepts/random-variables)
- [Try the interactive drill](https://learn.significanthobbies.com/drills/compute-expectation-variance)
- [All practice briefs](https://learn.significanthobbies.com/curriculum/drills/)