Fundamentals of Probability and Statistics for Engineers

It follows from Equations (6.34) and (6.36) that

In order to determine probability mass function based on the
assumptions stated above, let us first consider F igure 6.2 shows two
nonoverlapping intervals, [0, t) and In order that there are no
arrivals in the total interval we must have no arrivals in both
subintervals. Owing to the independence of arrivals in nonoverlapping inter-
vals, we thus can write

Rearranging Equation (6.38) and dividing both sides by gives

Upon letting we obtain the differential equation

Its solution satisfying the initial condition 1 is

The determination of p 1 (0,t) is similar. We first observe that one arrival in
can be accomplished only by having no arrival in subinterval [0,t)
and one arrival in or one arrival in [0,t) and no arrival in
Hence we have

0 t t+ t

No arrival No arrival

Figure6. 2 Interval[0,

SomeImportantDiscreteDistributions 175

p 0 …t;t‡t†ˆ 1 

X^1

kˆ 1

pk…t;t‡t†

ˆ 1 t‡o…t†: … 6 : 37 †

pk(0,t)

p 0 (0,t).

[t,t‡t).

[0,t‡t),

∆

t‡t)

p 0 … 0 ;t‡t†ˆp 0 … 0 ;t†p 0 …t;t‡t†

ˆp 0 … 0 ;t†‰ 1 t‡o…t†Š: … 6 : 38 †

p 0 … 0 ;t‡t†p 0 … 0 ;t†

t

ˆp 0 … 0 ;t†

o…t†

t



:

t!0,

dp 0 … 0 ;t†

dt

ˆp 0 … 0 ;t†: … 6 : 39 †

p 0 (0,0)ˆ

p 0 … 0 ;t†ˆet: … 6 : 40 †

[0,t‡t)
[t,t‡t), [t,t‡t).

t

p 1 … 0 ;t‡t†ˆp 0 … 0 ;t†p 1 …t;t‡t†‡p 1 … 0 ;t†p 0 …t;t‡t†: … 6 : 41 †