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(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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Parent category: Education