Russia 11 класс Probability and Statistics (basic)
Chapters: 2
1. Random variables
Expectation and variance · Law of large numbers
- Random Variables and Probability Distributions – A random variable X gives a number to every outcome of a chance experiment. A discrete random variable takes separate values; its probability distribution lists each value x with its probability P(X = x), and these add up to 1. The mean (expected value) is E(X) = Σ x·P(x). The variance Var(X) = E(X²) − [E(X)]² measures spread, and the standard deviation is its square root.
- 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
Normal 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).