Digital Communications

Appendix F: Probability and Random Variables

F.1 Random Variable

A random variable (rv) is unknown and unpredictable beforehand, that is, it has a random value, but its value is known completely once it occurs. A rv may be continuous or discrete. For example, the noise voltage generated by an electronic amplifier is random and has a continuous amplitude. On the other hand, coin flipping with outcomes head (H) and tail (T) is discrete. One does not know the outcome before flipping a coin. For each experiment, a value is assigned for each of the possible outcomes in the experiment. For example, the rv may be assumed to be X(s) = 1 for the outcome s = H and −1 for s = T. However, once the coin is flipped, the outcome (H or T) and hence X(s) is known. In the sequel we denote the rv simply by X but not as X(s).

A rv is characterized by its probability density function (pdf) or cumulative distribution function (cdf), which are interrelated. The cdf, F_X(x), of a rv X is defined by

(F.1)

which specifies the probability that the rv X is less than or equal to a real number x. The pdf of a rv X is defined as the derivative of the cdf:

(F.2)

Conversely, the cdf is defined by the integral of the pdf:

(F.3)

Based on (F.1)–(F.3), a rv has the following properties:

1. 0 ≤ F_X(x) ≤ 1 since and