Page 38 - Computational Statistics Handbook with MATLAB

24 Computational Statistics Handbook with MATLAB

2.5 Common Distributions

In this section, we provide a review of some useful probability distributions
and briefly describe some applications to modeling data. Most of these dis-
tributions are used in later chapters, so we take this opportunity to define
them and to fix our notation. We first cover two important discrete distribu-
tions: the binomial and the Poisson. These are followed by several continuous
distributions: the uniform, the normal, the exponential, the gamma, the chi-
square, the Weibull, the beta and the multivariate normal.

Binomia
l
BinomiaBinomia ll l
Binomia
Let’s say that we have an experiment, whose outcome can be labeled as a
‘success’ or a ‘failure’. If we let X = 1 denote a successful outcome and
X = 0 represent a failure, then we can write the probability mass function as
(
f 0() = PX = 0) = 1 – p,
(2.23)
(
f 1() = PX = 1) = p,
where p represents the probability of a successful outcome. A random vari-
able that follows the probability mass function in Equation 2.23 for 0 < p < 1
is called a Bernoulli random variable.
Now suppose we repeat this experiment for n trials, where each trial is
independent (the outcome from one trial does not influence the outcome of
another) and results in a success with probability p. If X denotes the number
of successes in these n trials, then X follows the binomial distribution with
parameters (n, p). Examples of binomial distributions with different parame-
ters are shown in Figure 2.3.
To calculate a binomial probability, we use the following formula:

n
,
,,
(
,
(
x
f xn p) = PX = x) =  p 1 –( p) n – x ; x = 0 1 … n . (2.24)
;

x
The mean and variance of a binomial distribution are given by
EX[] = np,
and
(
VX() = np 1 – p).

© 2002 by Chapman & Hall/CRC