Discrete Distribution Functions and Mathematical Expectation

Lesson 23 of STA301 marks the beginning of the Virtual University final term syllabus, exploring cumulative distribution functions and the foundational concept of Mathematical Expectation. The Cumulative Distribution Function (CDF) of a discrete random variable X, denoted F(x), represents the probability that the random variable takes on a value less than or equal to a specified real number x: F(x) = P(X <= x) = sum_{t <= x} f(t). The discrete CDF exhibits essential mathematical properties: it is a non-decreasing step function (staircase graph) with jump discontinuities occurring precisely at discrete support points, it is right-continuous, and it satisfies the asymptotic limits lim_{x -> -inf} F(x) = 0 and lim_{x -> +inf} F(x) = 1. The height of each step equals the point probability mass f(x_i). The lecture introduces Mathematical Expectation (Expected Value), denoted E(X) or mu. For a discrete random variable X with PMF f(x), the mathematical expectation is defined as the probability-weighted sum of all possible values: E(X) = sum [x * f(x)]. Physically analogous to the center of mass or centroid of a mass distribution, the expected value represents the long-run theoretical arithmetic average resulting from repeated experimental realizations. Expectation extends to functions of a random variable: E[g(X)] = sum [g(x) * f(x)]. Using this formulation, the Variance of X is derived as Var(X) = E[(X - mu)^2] = E(X^2) - [E(X)]^2. Furthermore, students analyze essential algebraic properties of expectation: the expected value of a constant is the constant itself, E(c) = c; expectation is a linear operator satisfying E(aX + b) = a*E(X) + b; and for variance, Var(c) = 0 while Var(aX + b) = a^2 * Var(X). The lecture also examines raw moments about zero (mu'_r = E(X^r)) and central moments about the mean (mu_r = E[(X - mu)^r]). In Virtual University final term examinations, verifying PMFs, computing E(X) and Var(X), and applying expectation linearity properties are high-yield computational problems. Practicing these problems using sta301 handouts and solved papers by Moaaz guarantees exam success.