Units 4-5 cover probability theory and distributions — the mathematical foundation for statistical inference. Understanding random variables and sampling distributions is essential for hypothesis testing.
P(A) between 0 and 1. Complement: P(A\') = 1 − P(A). Addition: P(A or B) = P(A) + P(B) − P(A and B). Mutually exclusive: P(A and B) = 0. Multiplication: P(A and B) = P(A) × P(B|A). Independent: P(A and B) = P(A) × P(B), and P(B|A) = P(B). Conditional: P(A|B) = P(A and B)/P(B). Tree diagrams and two-way tables for organisation.
Discrete random variable: list all outcomes and probabilities. Mean μ = Σ x·P(x). Variance σ² = Σ (x−μ)²·P(x). Binomial: X ~ B(n,p), fixed n trials, two outcomes, constant p, independent. P(X=k) = C(n,k)p^k(1-p)^(n-k). μ = np, σ = √(np(1-p)). Geometric: number of trials until first success. P(X=k) = (1-p)^(k-1)·p. μ = 1/p. Combining: μ(X+Y) = μX + μY. Var(X+Y) = Var(X) + Var(Y) if independent.
Sampling distribution: distribution of a statistic over all possible samples. CLT (Central Limit Theorem): for large n (≥30), sampling distribution of x̄ is approximately normal: N(μ, σ/√n). Sampling distribution of p̂: approximately N(p, √(p(1-p)/n)) when np ≥ 10 and n(1-p) ≥ 10. Larger samples → less variability.
The CLT states that regardless of the shape of the population distribution, the sampling distribution of the sample mean (x̄) approaches a normal distribution as the sample size n increases. Specifically: x̄ ~ N(μ, σ/√n) for large n (typically n ≥ 30). This is why the normal distribution is so important in statistics — it allows us to make inferences about population parameters even when the population is not normal. For proportions: p̂ ~ N(p, √(p(1-p)/n)) when both np and n(1-p) ≥ 10.
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