Question: 5.1.1 Let 1; 2; : : : be independent random variables, each having an exponential distribution with parameter . Define a new random variable X

5.1.1 Let 1; 2; : : : be independent random variables, each having an exponential distribution with parameter . Define a new random variable X as follows: If

1 > 1, then X D 0; if 1 1 but 1 C2 > 1, then set X D 1; in general, set X D k if

Show that X has a Poisson distribution with parameter . (Thus, the method outlined can be used to simulate a Poisson distribution.)
Hint: 1 C Ck has a gamma density

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Condition on 1 C Ck and use the law of total probability to show

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where F.x/ is the exponential distribution function.

1++k1

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