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== Problem 3: Random Chord == | == Problem 3: Random Chord == | ||
| − | A circle has radius <math>r</math>. Any chord of the circle is at distance at most <math>r</math> from the center. A random chord is drawn by first choosing its distance <math>D</math> from the center uniformly from the interval <math>[0,r]</math>, and then choosing any chord at that distance from the center. Find the PDF of <math>L</math>, the length of the chord. Draw a figure to illustrate. | + | A circle has radius <math>r</math>. Any [http://mathworld.wolfram.com/Chord.html chord] of the circle is at distance at most <math>r</math> from the center. A random chord is drawn by first choosing its distance <math>D</math> from the center uniformly from the interval <math>[0,r]</math>, and then choosing any chord at that distance from the center. Find the PDF of <math>L</math>, the length of the chord. Draw a figure to illustrate. |
== Problem 4: Fire Station == | == Problem 4: Fire Station == | ||
*(a) A fire station is to be located at a point <math>a</math> along a road of length <math>A</math>, <math>0 < A < \infty</math>. If fires will occur at points uniformly chosen on <math>(0,A)</math>, where should the station be located so as to minimize the expected distance from the fire? That is, choose <math>a</math> so as to minimize the quantity <math>E[|X - a|]</math> when <math>X</math> is uniformly distributed over <math>(0,A)</math>. | *(a) A fire station is to be located at a point <math>a</math> along a road of length <math>A</math>, <math>0 < A < \infty</math>. If fires will occur at points uniformly chosen on <math>(0,A)</math>, where should the station be located so as to minimize the expected distance from the fire? That is, choose <math>a</math> so as to minimize the quantity <math>E[|X - a|]</math> when <math>X</math> is uniformly distributed over <math>(0,A)</math>. | ||
*(b) Now suppose that the road is of infinite length--stretching from point 0 outward to <math>\infty</math>. If the distance of a fire from point 0 is exponentially distributed with rate <math>\lambda</math>, where should the fire station now be located? That is, we want to minimize <math>E[|X - a|]</math> with respect to <math>a</math> when <math>X</math> is now an exponential random variable with parameter <math>\lambda</math>. | *(b) Now suppose that the road is of infinite length--stretching from point 0 outward to <math>\infty</math>. If the distance of a fire from point 0 is exponentially distributed with rate <math>\lambda</math>, where should the fire station now be located? That is, we want to minimize <math>E[|X - a|]</math> with respect to <math>a</math> when <math>X</math> is now an exponential random variable with parameter <math>\lambda</math>. | ||
Revision as of 08:51, 30 September 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 $?
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 $.
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.
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 $.
