This exercise obtains a useful identity for the cdf of a Poisson cdf.

(a) Use Exercise 3.3.5 to show that this identity is true:

for λ > 0 and n a positive integer.

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Exercise 3.3.5

Show that

This demonstrates the relationship between the cdfs of the gamma and Poisson distributions.

(b) Obtain the identity used in Example 4.3.3, by making the transformation z = λx in the above integral.

Example 4.3.3