Summary of Probability and Statistics
I. Probability
Sample Space: Ω; Event: A Ω⊆
Basic Laws:
- 0 ≤ P(A) ≤ 1, P(Ω) = 1
- P(A ) = 1 - P(A)ᶜ
- P(A B) = P(A) + P(B) - P(A ∩ B)∪
Conditional Probability: P(A|B) = P(A ∩ B)/P(B)
Independence: P(A ∩ B) = P(A)P(B)
Law of Total Probability: P(A) = P(A|Bi)P(Bi)Σ
II. Random Variables and Distributions
Discrete: Probability Mass Function (PMF): f(x) = P(X = x)
Continuous: Probability Density Function (PDF): f(x) ≥ 0, ∫f(x)dx = 1
Cumulative Distribution Function (CDF): F(x) = P(X ≤ x)
Mean: E(X) = xP(x) (discrete), ∫x f(x) dx (continuous)Σ
Variance: Var(X) = E(X²) - [E(X)]²
III. Special Distributions
Binomial: B(n, p): P(X = k) = C(n,k)p^k(1-p)^(n-k); E(X) = np, Var(X) = np(1-p)
Poisson: P( ): P(X = k) = ^k * e^(- ) / k!; E(X) = Var(X) = λ λ λ λ
Uniform U(a,b): f(x) = 1/(b-a); E(X) = (a+b)/2, Var(X) = (b-a)^2/12
Exponential Exp( ): f(x) = e^(- x); E(X) = 1/ , Var(X) = 1/ ²λ λ λ λ λ
Normal N( , ²): f(x) = (1/√(2 ²)) e^(-(x- )²/2 ²)μ σ πσ μ σ
IV. Confidence Intervals
Known : CI = xA ± z /2 * /√nσ α σ
