Page 341 - Fundamentals of Probability and Statistics for Engineers

324 Fundamentals of Probability and Statistics for Engineers

Table 10.4 One-minute arrivals, for Example 10.3
Vehicles per minute (No.) Number of occurrences
0 0
1 0
2 1
3 3
4 5
5 7
6 13
7 12
8 8
9 9
10 13
11 10
12 5
13 6
14 4
15 5
16 4
17 0
18 1
nˆ 106

Answer: the hypothesized distribution is

x
e
p x†ˆ ; x ˆ 0; 1; 2; ... ; 10:11†
X
x!
where parameter needs to be estimated from the data. Thus, r ˆ 1.
To proceed, we first determine appropriate intervals A i such that n i 5 for
all i; these are shown in the first column of Table 10.5. Hence, kˆ 11.
The maximum likelihood estimate for is given by

n
1 X
^
ˆ x ˆ x j ˆ 9:09:
n
jˆ1
The substitution of this value for parameter in Equation (10.11) permits us to
calculate probabilities P(A i ) ˆ p i . For example,
4
X
p 1 ˆ p j†ˆ 0:052;
X
jˆ0
p 2 ˆ p 5†ˆ 0:058:
X

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