Other Important Random Variables - Random Signals and Processes Primer with MATLAB - page 333

Fig. 5.18 Relation between Poisson flow, exponential time, and constant intensity l

5.8 Geometric Random Variable

5.8.1 What Is a Geometric Random Variable and Where Does
This Name Come From?

Consider a Bernoulli experiment where the possible outcomes a re the events A
(success) with a probability of p and the complementary event A (failure) with a
probability q ¼ 1 p .
We are looking for t he probability that event A occurs in m th experiment; that is,
after ( m 1) failures ( A ).
Denoting this probability ( m , p ), we have:

Г

m 1

Gðm; pÞ¼ð 1 pÞ p: (5.181)

Similarly, one can find the probability of the first failure after m 1 successes
using a Bernoulli trials:

0

ðm; pÞ¼ð 1 pÞp m 1

G :

(5.182)

Note that the probabilities ( 5.181 ) and ( 5.182 ) are members of geometrical series
and that is why the name geometric is used.

5.8.2 Probability Distribution and Density Functions

Let X be a discrete random variable taking all possible values of m (from 1 to 1 .)
Therefore, the distribution function is

F X ðxÞ¼ 1

m¼ 1 Gðm; pÞuðx mÞ¼ 1

m 1

m¼ 1 ð 1 pÞ

puðx mÞ;

(5.183)

where Gðm; pÞ is the probability mass function given in ( 5.181 ).