Variance of a random variable, the geometric distribution and Chebyshev's inequality
Compute a variance from a distribution table, find the variance of a sum of independent variables, find a probability of 'until the first success' and bound a deviation with Chebyshev's inequality.
- France: grade 11
- Russia: grade 10
Guide
This studio follows the Russian advanced-level course 'Probability and statistics' (variance, properties of M and D, geometric and binomial distributions, Chebyshev's inequality and Bernoulli's theorem).
Variance measures how spread out the values are around the expected value: D(X) = M(X²) − (M(X))². For independent variables D(X + Y) = D(X) + D(Y), and a constant factor comes out squared: D(aX) = a²D(X). For the number of successes in n trials M(X) = np and D(X) = npq. If trials are repeated until the first success, P(X = k) = q^(k − 1) p. Chebyshev: P(|X − M| ≥ ε) ≤ D(X) ÷ ε².
Common mistakes: forgetting to subtract the square of the mean, not squaring the factor, and multiplying where variances should be added.