Probability And Statistical Methods For Engineers Worksheets With Answer Key - Mae 108, 2011 Page 4
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4(4) (10 points)
Earthquakes occur in a given region in accordance with a Poisson process with mean rate of 5 per
year.
(a) What is the probability that there will be at least two earthquakes in the first half of 2011?
Poisson process with ν = 5 and t = 1/2:
5 2
5 2
P (X
2) = 1 P (X
< 2) = 1 P (X
= 0) P (X
= 1) = 1 e
(5/2)e
71.3%
1 2
1 2
1 2
1 2
(b) What is the probability there will be no earthquake during the period including the last 6
months of 2010 and the first 9 months of 2011?
In this case we have a time t = (6 + 9)/12 = 15/12 = 5/4 year, and we want to have no occurence
hence the probability is
25 4
P (X
= 0) = e
0.2%
5 4
(c) [hard] Assuming that there are at least two earthquakes in the first half of 2011, what is the
probability that there will be at least four earthquakes over the first 9 months of 2011?
We use the multiplication rule:
P (X
4, X
2)
3 4
1 2
P (X
4 X
2) =
3 4
1 2
P (X
2)
1 2
The probability P (X
2) has been calculated in question (a). To get the probability P (X
1 2
3 4
4, X
2), we have to write the event as a sum of independent event. It is:
1 2
X
4, X
2 = X
= 2, X
2
X
= 3, X
1
X
4
3 4
1 2
1 2
1 4
1 2
1 4
1 2
and thus
2
5 2
5 4
5 4
3
5 2
5 4
P (X
4, X
2) = [(5/2)
e
/2!][1
e
(5/4)e
] + [(5/2)
e
/3!][1
e
]
3 4
1 2
5 2
5 2
2
5 2
3
5 2
+[1
e
(5/2)e
(5/2)
e
/2
(5/2)
e
/3!]
and thus we find P (X
4 X
2)
68.2%.
3 4
1 2
4
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