Fundamentals of Probability and Statistics for Engineers

To fix ideas in the following development, let us consider the problem of
passenger arrivals at a bus terminal during a specified time interval. We shall
use the notation X(0,t) to represent the number of arrivals during time interval
[0,t), where the notation [ ) denotes a left-closed and right-open interval; it is a
discrete random variable taking possible values 0, 1, 2,... , whose distribution
clearly depends on t. For clarity, its pmf is written as

to show its explicit dep endence on t. Note that this is different from our
standard notation for a pmf.
To study this problem, we make the following basic assumptions:
Assumption 1: the random variables
are mutually independent, that is, the numbers of passen-
ger arrivals in nonoverlapping time intervals are independent of each other.
Assumption 2: for sufficiently small

where stands for functions such that

This assumption says that, for a sufficiently small the probability of

having exactly one arrival is proportional to the length of The parameter

in Equation (6.34) is called the average density or mean rate of arrival for

reasons that will soon be made clear. F or simplicity, it is assumed to be a

constant in this discussion; however, there is no difficulty in allowing it to

vary with time.

Assumption 3: for sufficiently small

This condition implies that the probability of having two or more arrivals

during a sufficiently small interval is negligible.

174 Fundamentals of Probability and Statistics for Engineers

pk… 0 ;t†ˆP‰X… 0 ;t†ˆkŠ; kˆ 0 ; 1 ; 2 ;...; … 6 : 33 †

X(t 1 ,t 2 ), X(t 2 ,t 3 ),...,X(tn1,tn),

t 1 <t 2 <...<tn;

t,

.

.

p 1 …t;t‡t†ˆt‡o…t†… 6 : 34 †

o(t)

lim

t! 0

o…t†

t

ˆ 0 : … 6 : 35 †

t,

t.



. t,

X^1

kˆ 2

pk…t;t‡t†ˆo…t†… 6 : 36 †