Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

3.5 PROBABILITY DENSITY FUNCTION

The cdf is a probability measure for semi-open intervals of random variable X on the measurable space and is given by (3.10). Since we are interested in intervals, it is instructive to consider the underlying function for FX(x), denoted by the pdf FX(x), such that when it is integrated over , the cdf is obtained. We can compute the probability of an event given by a union of intervals as an integral of the underlying pdf over those intervals. The following theorem, which we provide without proof, states the existence of a pdf.

Theorem 3.2 (Radon–Nikodým). If the probability measure PX on the measurable event space is absolutely continuous, then for random variable X there exists nonnegative function FX(x) such that

(3.30)

for interval .

Since the probability measure is absolutely continuous, all combinations of inequalities can be used in (3.30), such that . Moreover, these can be expressed as

(3.31)

which follows from

(3.32)

The pdf is obtained by ...