Engineering Optimization: Theory and Practice, Fourth Edition

11.2 Basic Concepts of Probability Theory 643

11.2.8 Probability Distributions

There are several types of probability distributions (analytical models) for describ-

ing various types of discrete and continuous random variables. Some of the common

distributions are given below:

Discrete case Continuous case

Discrete uniform distribution Uniform distribution

Binomial Normal or Gaussian

Geometric Gamma

Multinomial Exponential

Poisson Beta

Hypergeometric Rayleigh

Negative binomial (or Pascal’s) Weibull

In any physical problem, one chooses a particular type of probability distribution

depending on (1) the nature of the problem, (2) the underlying assumptions associated

with the distribution, (3) the shape of the graph betweenf (x)orF (x)andxobtained

after plotting the available data, and (4) the convenience and simplicity afforded by the

distribution.

Normal Distribution. The best known and most widely used probability distribution

is the Gaussian or normal distribution. The normal distribution has a probability density

function given by

fX(x)=

1

√

2 π σX

e−^1 /^2 [(x−μX)/σX]

2

, −∞< x <∞ (11.45)

whereμXandσXare the parameters of the distribution, which are also the mea n and

standard deviation ofX, respectively. The normal distribution is often identified as

N(μX,σX).

Standard Normal Distribution. A normal distribution with parametersμX= and 0

σX= , called the 1 standard normal distribution, is denoted asN(0, 1). Thus the density

function of a standard normal variable (Z)is given by

fZ(z)=

1

√

2 π

e− z(

(^2) / 2 )
, −∞< z <∞ (11.46)
The distribution function of the standard normal variable (Z)is often designated as
φ(z)so that, with reference to Fig. 11.4,
φ(z 1 ) =p and z 1 =φ−^1 (p) (11.47)
wherepis the cumulative probability. The distribution functionN(0, 1) [i.e.,φ(z)] is
tabulated widely asstandard normal tables. For example, Table 11.1, gives the values
ofz,f (z), andφ(z)for positive values ofz. This is because the density function is
symmetric about the mean value (z=0) and hence
f (−z)=f (z) (11.48)
φ(−z)= 1 −φ(z) (11.49)