(→Problem 2: Minimum of Exponentials) |
|||
| (39 intermediate revisions by 24 users not shown) | |||
| Line 1: | Line 1: | ||
| + | [[Category:HW5]] | ||
| + | [[Category:ECE302Fall2008_ProfSanghavi]] | ||
| + | [[Category:probabilities]] | ||
| + | [[Category:ECE302]] | ||
| + | [[Category:homework]] | ||
| + | [[Category:problem solving]] | ||
| + | |||
== Instructions == | == Instructions == | ||
Homework 5 can be [https://engineering.purdue.edu/ece302/homeworks/HW5FA08.pdf downloaded here] on the [https://engineering.purdue.edu/ece302/ ECE 302 course website]. | Homework 5 can be [https://engineering.purdue.edu/ece302/homeworks/HW5FA08.pdf downloaded here] on the [https://engineering.purdue.edu/ece302/ ECE 302 course website]. | ||
| Line 8: | Line 15: | ||
[[5.1 - Katie Pekkarinen_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 | ||
| + | |||
| + | [[5.1 - Ben Carter_ECE302Fall2008sanghavi]] Regarding Spence | ||
| + | |||
| + | [[5.1 - Chris Cadwallader_ECE302Fall2008sanghavi]] Variance | ||
| + | |||
| + | [[5.1 - Junzhe Geng_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | [[5.1 - Allen Humphreys_ECE302Fall2008sanghavi]] TA's Help | ||
== Problem 2: Minimum of Exponentials == | == Problem 2: Minimum of Exponentials == | ||
*(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>. | ||
| + | |||
| + | |||
| + | [[5.2a Jared McNealis_ECE302Fall2008sanghavi]] comment by Anand Gautam | ||
| + | |||
| + | [[5.2b Shao-Fu Shih_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | [[5.2b Divyanshu Kamboj_ECE302Fall2008sanghavi]] | ||
== Problem 3: Random Chord == | == Problem 3: Random Chord == | ||
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. | 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. | ||
| + | |||
| + | |||
| + | [[5.3 - Katie Pekkarinen_ECE302Fall2008sanghavi]] | ||
| + | <br>[[5.3 - AJ Hartnett_ECE302Fall2008sanghavi]] Comment by Beau Morrison | ||
| + | |||
| + | [[5.3 - steve anderson_ECE302Fall2008sanghavi]] just how i thought to do this problem | ||
| + | |||
| + | [[5.3 - Zhongtian Wang_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | |||
| + | [[5.3 - Joe Gutierrez_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | [[5.3 - Suan-Aik Yeo_ECE302Fall2008sanghavi]] | ||
== Problem 4: Fire Station == | == Problem 4: Fire Station == | ||
| Line 20: | Line 67: | ||
*(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>. | ||
| − | [[ | + | [[5.4 Joon Young Kim_ECE302Fall2008sanghavi]] |
| + | |||
| + | [[5.4a Seraj Dosenbach_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | ==Tuesday TA Session Homework Notes== | ||
| + | [[HW5 TA Session Notes_ECE302Fall2008sanghavi|HW5 TA Session Notes]] - Allen Humphreys | ||
| + | ---- | ||
| + | [[Main_Page_ECE302Fall2008sanghavi|Back to ECE302 Fall 2008 Prof. Sanghavi]] | ||
Latest revision as of 11:56, 22 November 2011
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
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
5.1 - Ben Carter_ECE302Fall2008sanghavi Regarding Spence
5.1 - Chris Cadwallader_ECE302Fall2008sanghavi Variance
5.1 - Junzhe Geng_ECE302Fall2008sanghavi
5.1 - Allen Humphreys_ECE302Fall2008sanghavi TA's Help
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
5.2b Shao-Fu Shih_ECE302Fall2008sanghavi
5.2b Divyanshu Kamboj_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
5.3 - Zhongtian Wang_ECE302Fall2008sanghavi
5.3 - Joe Gutierrez_ECE302Fall2008sanghavi
5.3 - Suan-Aik Yeo_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 $.
5.4 Joon Young Kim_ECE302Fall2008sanghavi
5.4a Seraj Dosenbach_ECE302Fall2008sanghavi
Tuesday TA Session Homework Notes
HW5 TA Session Notes - Allen Humphreys
