core practice · coding problem
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.
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.
Try this drill in the practice workspace →
Hints when you need them
- E[X] = Σ x·P(X=x).
- Compute E[X²] before subtracting the squared mean.
Watch for these mistakes
- Confusing the PDF value with a probability (for continuous RVs)
- Assuming small samples look like their population distribution
- Using variance when you mean standard deviation in an explanation