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(→Problem 2: Minimum of Exponentials) |
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[[5.2b Shao-Fu Shih_ECE302Fall2008sanghavi]] | [[5.2b Shao-Fu Shih_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | [[5.2b Divyanshu Kamboj_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | From the memoryless property of Exponential Distribution function: | ||
| + | Suppose E1,λ and E1,μ are independent, then; | ||
| + | P[min{ E1,λ , E1,μ } > t] = P[E1,λ > t] . P[E1,μ } > t] | ||
| + | = eˉλt . eˉμt | ||
| + | = eˉ(λ + μ)t | ||
| + | which shows that minimum of E1,λ and E1,μ is exponentially distributed. | ||
| + | So, | ||
| + | E1, λ1+ λ2+ λ3+……. λn = min { E1,λ1, E1,λ2, E1,λ3, ……….., E1,λn } | ||
| + | Here, if we put λ = 1, then; | ||
| + | E1, 1+ 2+ 3+……. n = min { E1,1, E1,2, E1,3, ……….., E1,n } | ||
== Problem 3: Random Chord == | == Problem 3: Random Chord == | ||
Revision as of 17:39, 6 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
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 - Junzhe Geng_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 $.
5.2a Jared McNealis_ECE302Fall2008sanghavi comment by Anand Gautam
5.2b Shao-Fu Shih_ECE302Fall2008sanghavi
5.2b Divyanshu Kamboj_ECE302Fall2008sanghavi
From the memoryless property of Exponential Distribution function: Suppose E1,λ and E1,μ are independent, then; P[min{ E1,λ , E1,μ } > t] = P[E1,λ > t] . P[E1,μ } > t] = eˉλt . eˉμt = eˉ(λ + μ)t which shows that minimum of E1,λ and E1,μ is exponentially distributed. So, E1, λ1+ λ2+ λ3+……. λn = min { E1,λ1, E1,λ2, E1,λ3, ……….., E1,λn } Here, if we put λ = 1, then; E1, 1+ 2+ 3+……. n = min { E1,1, E1,2, E1,3, ……….., E1,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.
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
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 $.
