# Segment 9. Characteristic Functions

## Contents

#### Watch this segment

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Links to the slides: PDF file or PowerPoint file

### Problems

#### To Calculate

1. Use characteristic functions to show that the sum of two independent Gaussian random variables is itself a Gaussian random variable. What is its mean and variance?

2. Calculate (don't just look up) the characteristic function of the Exponential distribution.

#### To Think About

1. Learn enough about contour integration to be able to make sense of Saul's explanation at the bottom of slide 7. Then draw a picture of the contours, label the pole(s), and show how you calculate their residues.

2. Do you think that characteristic functions are ever useful computationally (that is, not just analytically to prove theorems)?

## Class Activity

Alternate to problem 6(a): Show that a process with constant-rate exponentially distributed waiting times between events is a Poisson process (that has a Poisson distribution for the number of events in any fixed time interval).