Define random variables; properties

Random variable

A random variable X is a function that assigns a real number to each outcome of a random experiment. Example: the number shown on a die, the inter-arrival time of customers, or the service time at a bank counter.

• Discrete random variable: takes countable values (number of arrivals in an hour). Described by a probability mass function p(x) = P(X = x).
• Continuous random variable: takes any value in an interval (service time). Described by a probability density function f(x), with P(a ≤ X ≤ b) = ∫ f(x) dx from a to b.

Properties of random variables

1. Probability bounds. 0 ≤ p(x) ≤ 1 for discrete X, and f(x) ≥ 0 for continuous X.
2. Total probability is 1. Σ p(x) = 1 (discrete), ∫ f(x) dx = 1 over the whole range (continuous).
3. Cumulative distribution function. F(x) = P(X ≤ x). It is non-decreasing, with F(−∞) = 0 and F(+∞) = 1.
4. Expected value (mean). E(X) = Σ x·p(x) or ∫ x·f(x) dx. It is the long-run average. Also E(aX + b) = aE(X) + b, and E(X + Y) = E(X) + E(Y).
5. Variance. V(X) = E(X²) − [E(X)]². The standard deviation is σ = √V(X). Also V(aX + b) = a²V(X).
6. Independence. X and Y are independent if P(X = x, Y = y) = P(X = x)·P(Y = y). Then E(XY) = E(X)E(Y) and V(X + Y) = V(X) + V(Y).

For example, the random number R uniform on [0, 1] has f(R) = 1, E(R) = 1/2 and V(R) = 1/12.