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*(a) <math>X_1</math> is an exponential random variable with parameter <math>\lambda_1</math>, and <math>X_2</math> with <math>\lambda_2</math>. Let <math>Y = \min(X_1,X_2)</math>. What is the PDF of <math>Y</math>? Is <math>Y</math> one of the common random variables? | *(a) <math>X_1</math> is an exponential random variable with parameter <math>\lambda_1</math>, and <math>X_2</math> with <math>\lambda_2</math>. Let <math>Y = \min(X_1,X_2)</math>. What is the PDF of <math>Y</math>? Is <math>Y</math> one of the common random variables? | ||
*(b) Use induction to show that the minimum of <math>n</math> exponential random variables with parameter 1 is an exponential random variable with paramter <math>n</math>. | *(b) Use induction to show that the minimum of <math>n</math> exponential random variables with parameter 1 is an exponential random variable with paramter <math>n</math>. | ||
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| + | [[5.2 Jared McNealis_ECE302Fall2008sanghavi]] | ||
== Problem 3: Random Chord == | == Problem 3: Random Chord == | ||
Revision as of 08:41, 5 October 2008
Contents
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
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 $.
5.2 Jared McNealis_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
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 $.
