Continuous Random Variables and Density Functions

Lesson 24 of STA301 transitions from discrete systems to Continuous Random Variables and Probability Density Functions (PDF). A continuous random variable X can take on any real numerical value within a continuum or continuous interval, representing physical measurements such as time, weight, length, or voltage. Because the number of possible points in any real interval is uncountably infinite, the probability of a continuous variable assuming any single exact isolated value is mathematically zero: P(X = c) = 0 for any specific constant c. Consequently, non-zero probabilities are defined exclusively over intervals, where the inclusion or exclusion of boundary endpoints does not alter the probability: P(a <= X <= b) = P(a < X <= b) = P(a <= X < b) = P(a < X < b). The probability behavior of a continuous variable is governed by its Probability Density Function f(x). Graphically, the probability that X falls in the interval [a, b] corresponds to the definite integral representing the area under the curve y = f(x) between x = a and x = b: P(a <= X <= b) = integral_a^b f(x) dx. Any valid continuous PDF must satisfy two fundamental axioms: first, f(x) >= 0 for all x in its domain; second, the total area under the entire density curve over all real space equals unity: integral_{-inf}^{+inf} f(x) dx = 1. The lecture details the Continuous Cumulative Distribution Function F(x), defined as F(x) = P(X <= x) = integral_{-inf}^x f(t) dt. Unlike its discrete counterpart, the continuous CDF is a smooth, continuous, non-decreasing S-shaped curve ranging from 0 to 1. By the Fundamental Theorem of Calculus, the density function is recovered by differentiating the distribution function: f(x) = d/dx [F(x)]. In Virtual University final term evaluations, integrating density functions to find unknown normalizing constants, deriving CDFs, and evaluating interval probabilities are frequent subjective questions. Diligent revision of sta301 handouts and solved past papers ensures mastery.