Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, 3rd Edition

4

Continuous Random Variables

4.1   Continuous Sample Space

A random variable X is continuous if the range S_X consists of one or more intervals. For each x ∊ S_X, P[X = x] = 0.

Until now, we have studied discrete random variables. By definition, the range of a discrete random variable is a countable set of numbers. This chapter analyzes random variables that range over continuous sets of numbers. A continuous set of numbers, sometimes referred to as an interval, contains all of the real numbers between two limits. Many experiments lead to random variables with a range that is a continuous interval. Examples include measuring T, the arrival time of a particle (S_T = {t|0 ≤ t < ∞}); measuring V, the voltage across a resistor (S_V = {v| − ∞ < v < ∞}); and measuring the phase angle A of a sinusoidal radio wave (S_A = {a|0 ≤ a < 2π}). We will call T, V, and A continuous random variables, although we will defer a formal definition until Section 4.2.

Consistent with the axioms of probability, we assign numbers between zero and one to all events (sets of elements) in the sample space. A distinguishing feature of the models of continuous random variables is that the probability of each individual outcome is zero! To understand this intuitively, consider an experiment in which the observation is the arrival time of the professor at a class. Assume this professor always arrives between 8:55 and 9:05. We model the arrival time as a random variable T minutes relative to 9:00 o’clock. Therefore, ...