# Moment generating function and poisson distribution

** Please see the attached file for the complete solution **

Shirley runs a real estate company. She counts the total number of flats that she sells everyday. Let X_t be the total number of flats she sells on day t (t=1,2,...,7), and let X be the total number of flats she sells for the week. Suppose X_t are independent and identically distributed Poisson distribution with expectation theta.

(Please see the attached file)

(a) Find the moment generating functions for X_1 and X

(b) Show X has a Poisson distribution with mean 7*theta

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#### Solution Preview

Please refer to the pdf attached.

a) Let us recall the definition for the moment generating function for a random variable

(please see the attached file) With this in mind, let us find the moment generating function for X_1 first. By definition, since X_1 can only take values from 0, 1, ...

#### Solution Summary

This solution provides a step-by-step calculation of the moment generating function, and illustrates how to use the moment generating function to determine the probability distribution.