Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.9 IMPORTANT CONTINUOUS DISTRIBUTIONS

0

0

0. 2

0. 4

0. 6

0. 8

1

1

2 3 4

y

g(y)

μ=0,σ=0

μ=0,σ=0. 5

μ=0,σ=1. 5

μ=1,σ=1

Figure 30.15 The PDFg(y) for the log-normal distribution for various values

of the parametersμandσ.

The probability that an event occurs in the next infinitestimal interval [x, x+dx]

is given byλdx,sothat

Pr(the first event occurs in interval [x, x+dx]) =e−λxλdx.

Hence the required probability density function is given by

f(x)=λe−λx.

The expectation and variance of the exponential distribution can be evaluated as

1 /λand (1/λ)^2 respectively. The MGF is given by

M(t)=

λ

λ−t

. (30.117)

We may generalise the above discussion to obtain the PDF for the interval

between everyrth event in a Poisson process or, equivalently, the interval (waiting

time) before therth event. We begin by using the Poisson distribution to give

Pr(r−1 events occur in intervalx)=e−λx

(λx)r−^1

(r−1)!

,

from which we obtain

Pr(rth event occurs in the interval [x, x+dx]) =e−λx

(λx)r−^1

(r−1)!

λdx.

Thus the required PDF is

f(x)=

λ

(r−1)!

(λx)r−^1 e−λx, (30.118)

which is known as thegamma distributionof orderrwith parameterλ. Although

our derivation applies only whenris a positive integer, the gamma distribution is