Integral

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.

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