Fundamentals of Probability and Statistics for Engineers

Substituting Equations (6.34), (6.37), and (6.40) into Equation (6.41) and
letting we obtain

which yields

Continuing in this way we find, for the general term,

Equation (6. 44 ) gives the pmf of X(0,t), the number of arrivals during
time interval [ 0 ,t) subject to the assumptions stated above. It is called the
Poisson distribution, with parameters and t. However, since and t appear in
Equation (6. 44 ) as a product, t, it can be replaced by a single parameter ,
andsowecanalsowrite

Themeanof isgivenby

Similarly,wecanshowthat

ItisseenfromEquation( 6. 46 )thatparameter isequaltotheaverage
numberofarrivalsperunitintervaloftime;thename‘meanrateofarrival’for
,asmentionedearlier,isthusjustified.Indeterminingthevalueofthis
parameterinagivenproblem,itcanbeestimatedfromobservationsby

176 FundamentalsofProbabilityandStatisticsforEngineers

t! 0

dp 1 … 0 ;t†

dt

ˆp 1 … 0 ;t†‡et; p 1 … 0 ; 0 †ˆ 0 ; … 6 : 42 †

p 1 … 0 ;t†ˆtet: … 6 : 43 †

pk… 0 ;t†ˆ

…t†ket

k!

; kˆ 0 ; 1 ; 2 ;...: … 6 : 44 †

 

 

ˆt,

pk… 0 ;t†ˆ

ke

k!

; kˆ 0 ; 1 ; 2 ;...: … 6 : 45 †

X(0,t)

EfX… 0 ;t†gˆ

X^1

kˆ 0

kpk… 0 ;t†ˆet

X^1

kˆ 0

k…t†k

k!

ˆtet

X^1

kˆ 1

…t†k^1

…k 1 †!

ˆtetetˆt:

… 6 : 46 †

^2 X… 0 ;t†ˆt: … 6 : 47 †





m/n,