Discrete random variables

Probability distribution

• A discrete random variable X takes countable values x₁, x₂, …, with probabilities p₁, p₂, … summing to 1.
• Example: X = number of heads in two tosses → P(0) = ¼, P(1) = ½, P(2) = ¼, and ¼ + ½ + ¼ = 1.

Expectation and variance

• E(X) = Σ x·P(x) — the mean of the distribution.
• Var(X) = Σ x²·P(x) − [E(X)]², and the standard deviation is √Var(X).
• For the two-toss example: E(X) = 0(¼) + 1(½) + 2(¼) = 1; E(X²) = 0 + ½ + 1 = 1.5; Var(X) = 1.5 − 1 = 0.5; σ ≈ 0.707.
• Properties: E(aX + b) = aE(X) + b and Var(aX + b) = a²Var(X).