Suppose that the number of years, Xi , for i = 1, 2, . . . between earthquakes in a certain

region are independently and identically distributed with probability density function,

fX(x) = (

1

10

e

x/10 ,

for

x

0

0

,

for

x <

0

(i) What is the probability that the time between any two earthquakes will be between 8

and 12 years?

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(ii) Seismologists in this region start monitoring the occurrence of earthquakes on January

1, 2010. If no earthquake has occurred by January 1, 2021, what is the probability that

there will be an earthquake in between January 1, 2021 and January 1, 2031?

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(iii) What is the moment generating function (mgf) of X? A derivation is not required although

you must state any result you use.

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(iv) Refer to Part (ii). Suppose that an earthquake occurs on January 1, 2010. Derive the mgf

of the time to when the 5th earthquake after January 1, 2010 occurs. Justify your steps.

What is the name of the distribution that has this mgf ?

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(v) Refer to Part (iv). Could there be another distribution with this mgf ? Why or why not?

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