Page 193 - Fundamentals of Probability and Statistics for Engineers

176 Fundamentals of Probability and Statistics for Engineers

Substituting Equations (6.34), (6.37), and (6.40) into Equation (6.41) and
letting t ! 0 we obtain

dp 0; t† t
1
ˆ p 0; t†‡ e ; p 0; 0†ˆ 0; 6:42†
1
1
dt
which yields
p 0; t†ˆ te t : 6:43†
1
Continuing in this way we find, for the general term,

k t
t† e
p 0; t†ˆ ; k ˆ 0; 1; 2; ... : 6:44†
k
k!
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 ,
ˆ t, and so we can also write

k
e
p 0; t†ˆ ; k ˆ 0; 1; 2; ... : 6:45†
k
k!
The mean of X(0, t) is given by

1 1 k
X t X k t†
EfX 0; t†g ˆ kp 0; t†ˆ e
k
kˆ0 kˆ0 k! 6:46†
1 k 1
X t†
ˆ te t ˆ te t t
e ˆ t:
k 1†!
kˆ1
Similarly, we can show that
2 ˆ t: 6:47†
X 0;t†

It is seen from Equation (6.46) that parameter is equal to the average
number of arrivals per unit interval of time; the name ‘mean rate of arrival’ for
, as mentioned earlier, is thus justified. In determining the value of this
parameter in a given problem, it can be estimated from observations by m/n,

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