# Difference between revisions of "Eleisha's Segment 9: Characteristic Functions"

(Created page with "<b>To Calculate: </b> 1. Use characteristic functions to show that the sum of two independent Gaussian random variables is itself a Gaussian random variable. What is its mea...") |
|||

Line 2: | Line 2: | ||

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

+ | |||

+ | <math> \psi_X(t) = \int_0^\infty e^{itx} \cdot \lambda e^{\lambda x} dx </math> | ||

+ | |||

<b>To Think About: </b> | <b>To Think About: </b> |

## Revision as of 13:48, 22 February 2014

**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.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_X(t) = \int_0^\infty e^{itx} \cdot \lambda e^{\lambda x} dx }**

**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)?

**Back to** Eleisha Jackson