Standard Probability Distributions - Advanced Probability Theory for Biomedical Engineers - page 19

so that

F t r ( τ ) − F t r ( τ − h ) o ( h )

h

= λ p ( r − 1 ,τ − h ) + .

h

Taking the limit as h 0 we find that the PDF for the r th order interarrival time, that is, the

time interval from any starting point to the r th success after it, is

→

r r − 1 e − λτ

= λ

τ

f t r ( τ ) u ( τ ) , r = 1 , 2 ,.... (5.54)

( r − 1)!

This PDF is known as the Erlang PDF. Clearly, with r = 1, we have the exponential PDF:

e − λτ u ( f t ( τ ) = λ τ ) . (5.55)

The RV t is called the first-order interarrival time.
The Erlang PDF is a special case of the gamma PDF:

r r − 1 e − λα

= λ

α

f x ( α ) u ( α ) , (5.56)

( r )

for any real r > 0, λ> 0, where is the gamma function

∞

r − 1 e − α d

( r ) = α α. (5.57)

0

Straightforward integration reveals that (1) = 1 and ( r + 1) = r ( r ) so that if r is a positive

integer then 1)!—for this reason the gamma function is often called the factorial

function. Using the above definition for

( r ) = ( r −

( r ), it is easily shown that the moment generating

function for a gamma-distributed RV is

r

λ
λ − η

M x (

η ) = , for η<λ.

(5.58)

The characteristic function is thus

r

λ
λ −

φ x ( t ) = . (5.59)

jt

It follows that the mean and variance are

r
λ ,

r
λ

2
x

η = and σ = . (5.60) x

2

Figure 5.8 illustrates the PDF and magnitude of the characteristic function for a RV with
gamma distribution with r = 3 and λ = 0 . 2.