Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

The above results may be extended. For example, if the random variables

Xi,i=1, 2 ,...,n, are distributed asXi∼N(μi,σ^2 i) then the random variable

Z=

∑

iciXi(where theciare constants) is distributed asZ∼N(

∑

iciμi,

∑

ic

2

iσ

2

i).

30.9.2 The log-normal distribution

If the random variableXfollows a Gaussian distribution then the variable

Y=eXis described by alog-normaldistribution. Clearly, ifXcan take values

in the range−∞to∞,thenYwill lie between 0 and∞. The probability density

function forYis found using the result (30.58). It is

g(y)=f(x(y))

∣

∣

∣

∣

dx

dy

∣

∣

∣

∣=

1

σ

√

2 π

1

y

exp

[

−

(lny−μ)^2

2 σ^2

]

.

We note that μandσ^2 are not the mean and variance of the log-normal

distribution, but rather the parameters of the corresponding Gaussian distribution

forX. The mean and variance ofY, however, can be found straightforwardly

using the MGF ofX, which readsMX(t)=E[etX]=exp(μt+^12 σ^2 t^2 ). Thus, the

mean ofYis given by

E[Y]=E[eX]=MX(1) = exp(μ+^12 σ^2 ),

and the variance ofYreads

V[Y]=E[Y^2 ]−(E[Y])^2 =E[e^2 X]−(E[eX])^2

=MX(2)−[MX(1)]^2 =exp(2μ+σ^2 )[exp(σ^2 )−1].

In figure 30.15, we plot some examples of the log-normal distribution for various

values of the parametersμandσ^2.

30.9.3 The exponential and gamma distributions

The exponential distribution with positive parameterλis given by

f(x)=

{

λe−λx forx> 0 ,

0forx≤ 0

(30.116)

and satisfies

∫∞
−∞f(x)dx= 1 as required. The exponential distribution occurs nat-
urally if we consider the distribution of the length of intervals between successive

events in a Poisson process or, equivalently, the distribution of the interval (i.e.

the waiting time) before the first event. If the average number of events per unit

interval isλthen on average there areλxevents in intervalx, so that from the

Poisson distribution the probability that there will be no events in this interval is

given by

Pr(no events in intervalx)=e−λx.