Russia 11 класс Probability and Statistics (advanced)
Chapters: 3
1. Law of large numbers
Chebyshev and Bernoulli
- The Law of Large Numbers – If you repeat the same random experiment many times, the relative frequency of an event gets closer and closer to its probability p. More precisely, for any small distance ε, the chance that the frequency (or the sample mean) is farther than ε from the true value goes to 0 as the number of trials n grows. Chebyshev's inequality gives a bound: P(|X̄ − μ| ≥ ε) ≤ σ² ÷ (nε²).
2. Continuous random variables
Densities · Poisson distribution
- Normal Distribution – A normal distribution is a continuous, symmetric, bell-shaped distribution fixed by its mean μ (centre) and standard deviation σ (spread): X ~ N(μ, σ²). Mean = median = mode. About 68% of values lie within 1σ of μ, 95% within 2σ and 99.7% within 3σ. Any normal value is turned into a standard score z = (x − μ)/σ, which follows N(0, 1); probabilities are areas under the curve, read from a table or calculator as Φ(z).
3. Correlation
Correlation and regression
- Correlation: Scatter Diagram, Karl Pearson's Coefficient and Spearman's Rank Correlation – Correlation tells how two variables move together. It is positive when both rise together, negative when one rises as the other falls, and zero when there is no straight-line pattern. A scatter diagram shows it as a picture. Karl Pearson's coefficient r measures its direction and strength and always lies between −1 and +1. Spearman's rank correlation R uses ranks and works for qualities like beauty or honesty; tied ranks need a small correction.