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236 Chapter 4 Special Distributions
Proof Suppose an event has occurred at time a. Consider the interval that extends
from a to a + y. Since the (Poisson) events are occurring at the rate of λ per unit time,
−λy
0
the probability that no outcomes will occur in the interval (a, a + y) is e 0!(λy) = e−λy .
Define the random variable Y to denote the interval between consecutive occurrences. Notice that there will be no occurrences in the interval (a, a + y) only if
Y > y. Therefore,
P(Y > y) = e−λy
or, equivalently,
FY (y) = P(Y ≤ y) = 1 − P(Y > y) = 1 − e−λy
Let f Y (y) be the (unknown) pdf for Y . It must be true that
% y
P(Y ≤ y) =
f Y (t) dt
0
Taking derivatives of the two expressions for FY (y) gives
% y
d
d
(1 − e−λy )
f Y (t) dt =
dy 0
dy
which implies that
f Y (y) = λe−λy ,
y >0
Case Study 4.2.4
Over “short” geological periods, a volcano’s eruptions are believed to be
Poisson events—that is, they are thought to occur independently and at a constant rate. If so, the pdf describing the intervals between eruptions should
have the form f Y (y) = λe−λy . Collected for the purpose of testing that presumption are the data in Table 4.2.5, showing the intervals (in months) that
elapsed between thirty-seven consecutive eruptions of Mauna Loa, a fourteenthousand-foot volcano in Hawaii (106). During the period covered—1832 to
1950—eruptions were occurring at the rate of λ = 0.027 per month (or once
every 3.1 years). Is the variability in these thirty-six yi ’s consistent with the
statement of Theorem 4.2.3?
Table 4.2.5
126
73
26
6
41
26
73
23
21
18
11
3
3
2
6
6
12
38
6
65
68
41
38
50
37
94
16
40
77
91
23
51
20
18
61
12
To answer that question requires that the data be reduced to a densityscaled histogram and superimposed on a graph of the predicted exponential pdf
(Continued on next page)