Difference between revisions of "Homework5 ECE302Fall2008sanghavi" - Rhea

Revision as of 15:22, 6 October 2008

Instructions

Homework 5 can be downloaded here on the ECE 302 course website.

Problem 1: Coupon Collector

Each brand of candy bar has one coupon in it. There are $ n $ different coupons in total; getting at least one coupon of each type entitles you to a prize. Each candy bar you eat can have any one of the coupons in it, with all being equally likely. Let $ X $ be the (random) number of candy bars you eat before you have all coupons. What are the mean and variance of $ X $?

5.1 - Henry Michl_ECE302Fall2008sanghavi

5.1 - Katie Pekkarinen_ECE302Fall2008sanghavi

5.1 - Ben Wurtz_ECE302Fall2008sanghavi

5.1 - Jayanth Athreya_ECE302Fall2008sanghavi

5.1 - Virgil Hsieh_ECE302Fall2008sanghavi Note to Henry, Katie, Ben, and Jayanth

5.1 - Brian Thomas_ECE302Fall2008sanghavi Note to Virgil, expansion of Henry's idea

5.1 - Virgil Hsieh 2_ECE302Fall2008sanghavi Expansion on Brian's idea

5.1 - Spencer Mitchell_ECE302Fall2008sanghavi Expansion on Virgil's idea

Problem 2: Minimum of Exponentials

• (a) $ X_1 $ is an exponential random variable with parameter $ \lambda_1 $, and $ X_2 $ with $ \lambda_2 $. Let $ Y = \min(X_1,X_2) $. What is the PDF of $ Y $? Is $ Y $ one of the common random variables?
• (b) Use induction to show that the minimum of $ n $ exponential random variables with parameter 1 is an exponential random variable with paramter $ n $.

5.2a Jared McNealis_ECE302Fall2008sanghavi comment by Anand Gautam

5.2a Christopher Wacnik_ECE302Fall2008sanghavi

5.2b Shao-Fu Shih_ECE302Fall2008sanghavi

Problem 3: Random Chord

A circle has radius $ r $. Any chord of the circle is at distance at most $ r $ from the center. A random chord is drawn by first choosing its distance $ D $ from the center uniformly from the interval $ [0,r] $, and then choosing any chord at that distance from the center. Find the PDF of $ L $, the length of the chord. Draw a figure to illustrate.

5.3 - Katie Pekkarinen_ECE302Fall2008sanghavi
5.3 - AJ Hartnett_ECE302Fall2008sanghavi Comment by Beau Morrison

5.3 - steve anderson_ECE302Fall2008sanghavi just how i thought to do this problem

Problem 4: Fire Station

• (a) A fire station is to be located at a point $ a $ along a road of length $ A $, $ 0 < A < \infty $. If fires will occur at points uniformly chosen on $ (0,A) $, where should the station be located so as to minimize the expected distance from the fire? That is, choose $ a $ so as to minimize the quantity $ E[|X - a|] $ when $ X $ is uniformly distributed over $ (0,A) $.
• (b) Now suppose that the road is of infinite length--stretching from point 0 outward to $ \infty $. If the distance of a fire from point 0 is exponentially distributed with rate $ \lambda $, where should the fire station now be located? That is, we want to minimize $ E[|X - a|] $ with respect to $ a $ when $ X $ is now an exponential random variable with parameter $ \lambda $.

5.4 Joon Young Kim_ECE302Fall2008sanghavi

5.4a Seraj Dosenbach_ECE302Fall2008sanghavi