Standard Probability Distributions - Advanced Probability Theory for Biomedical Engineers - page 7

bit, or an even and odd number in a die toss. Let us call one of the events a success, the other
a failure. The Bernoulli PMF describes the probability of k successes in n trials of a Bernoulli
experiment. The first two chapters used this PMF repeatedly in problems dealing with games
of chance and in situations where there were only two possible outcomes in any given trial.
For biomedical engineers, the Bernoulli distribution is used in infectious disease problems and
other applications. The Bernoulli distribution is also known as a Binomial distribution.

Definition 5.3.1. A discrete RV x is Bernoulli distributed if the PMF for x is

⎧
⎨
⎩

n
k

p k q n − k

, k = 0 , 1 ,..., n

p x ( k ) = (5.19)

0 , otherwise ,

where p = probability of success and q = 1 − p .

The characteristic function can be found using the binomial theorem:

n
k

( pe jt ) k q n − k

n

pe jt ) n φ

x ( t )

= = ( q + . (5.20)

k = 0

Figure 5.5 illustrates the PMF and the characteristic function magnitude for a discrete RV with
Bernoulli distribution, p 30.

Using the moment generating property of characteristic functions, the mean and variance
of a Bernoulli RV can be shown to be

= 0 . 2, and n =

2
x

η = np , σ = npq . (5.21)

x

α

φ

p (

)

x (t)

1

.18

.09

α t

0 10 20 0 1 2

(a) (b)

FIGURE 5.5: (a) PMF and (b) characteristic function magnitude for discrete RV with Bernoulli distri-
bution, p = 0 . 2 and n = 30.