Probability Distributions — Formula Sheet

Two conditions for a valid PMF

(i) f(xₖ) ≥ 0 for all k, and (ii) Σf(xₖ) = 1

For continuous X

P(X = a) = 0 for every single value a — probability at a point is ZERO

The Probability Density Function (PDF) f(x) satisfies

(i) f(x) ≥ 0 and (ii) ∫_{-∞}^{∞} f(x)dx = 1

For a discrete random variable X taking values x₁, x₂, x₃, ..., the PMF is f(x) = P(X = x).

For discrete variables

F(x) = Σ[xᵢ ≤ x] f(xᵢ).