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More Examples, Poisson Distribution
Example 1
Suppose that the number
X
of tornados observed in a particular region during a oneyear period
has a Poisson distribution with
¸
= 8
.
a.
Compute
P
(
X
·
5)
,
P
(6
·
X
·
9)
, and
P
(10
·
X
)
.
Solution
The desired probabilities are:
P
(
X
·
5)
=
5
X
x
=0
exp(
¡
8)8
x
x
!
= 0
:
1912
P
(6
·
X
·
9)
=
9
X
x
=6
exp(
¡
8)8
x
x
!
= 0
:
5254
P
(10
·
X
)
=
1
X
x
=10
exp(
¡
8)8
x
x
!
= 0
:
2834
:
b.
How many tornados can be expected to be observed during the oneyear period and what is the
standard deviation of the number of observed tornados?
Solution
E
(
X
) =
¸
= 8
,
¾
X
=
p
¸
=
p
8 = 2
:
8284
:
Example 2
Consider writing onto a computer disk and then sending it through a certi…er that
counts the number of missing pulses. Suppose this number
X
has a Poisson distribution with
¸
= 0
:
2
.
a.
What is the probability that a disk has exactly one missing pulse?
Solution
P
(
X
= 1) =
exp(
¡
0
:
2)0
:
2
1
1!
= 0
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 Fall '10
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