(→Problem 1: Ceiling of an Exponential) |
|||
| (47 intermediate revisions by 24 users not shown) | |||
| Line 1: | Line 1: | ||
| + | [[Category:ECE302Fall2008_ProfSanghavi]] | ||
| + | [[Category:probabilities]] | ||
| + | [[Category:ECE302]] | ||
| + | [[Category:homework]] | ||
| + | [[Category:problem solving]] | ||
| + | |||
== Instructions == | == Instructions == | ||
Homework 6 can be [https://engineering.purdue.edu/ece302/homeworks/HW6FA08.pdf downloaded here] on the [https://engineering.purdue.edu/ece302/ ECE 302 course website]. | Homework 6 can be [https://engineering.purdue.edu/ece302/homeworks/HW6FA08.pdf downloaded here] on the [https://engineering.purdue.edu/ece302/ ECE 302 course website]. | ||
| Line 6: | Line 12: | ||
<math>\mathrm{ceil}(a)</math> = <math>a</math> if <math>a</math> is an integer | <math>\mathrm{ceil}(a)</math> = <math>a</math> if <math>a</math> is an integer | ||
| − | + | = the smallest integer bigger than <math>a</math> if <math>a</math> is not an integer | |
What is the PMF of <math>Y</math>? Is it one of the common random variables? (Hint: for all <math>k</math>, find the quantity <math>P(Y > k)</math>. Then find the PMF) | What is the PMF of <math>Y</math>? Is it one of the common random variables? (Hint: for all <math>k</math>, find the quantity <math>P(Y > k)</math>. Then find the PMF) | ||
| + | |||
| + | *[[Tiffany Sukwanto 6.1_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | *[[Joshua Long 6.1_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | *[[Justin Mauck 6.1_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | == Problem 2: Fair Wages == | ||
| + | ``I do not have problems with anyone earning above average, as long as no one earns below average." - a quote (mistakenly attributed to) Max Weber. Can such a situation occur? Justify your answer. | ||
| + | |||
| + | *[[Brian Thomas 6.2_ECE302Fall2008sanghavi]] One possible solution | ||
| + | *[[Gregory Pajot 6.2_ECE302Fall2008sanghavi]] | ||
| + | *[[Virgil Hsieh 6.2_ECE302Fall2008sanghavi]] | ||
| + | *[[Zhongtian Wang 6.2_ECE302Fall2008sanghavi]] | ||
| + | *[[Michael Allen 6.2_ECE302Fall2008sanghavi]] | ||
| + | *[[Christopher Wacnik 6.2_ECE302Fall2008sanghavi]] | ||
| + | *[[Sahil Khosla 6.2_ECE302Fall2008sanghavi]] | ||
| + | *[[AJ Hartnett 6.2 --Different answer than above!_ECE302Fall2008sanghavi]] | ||
| + | *[[Jaewoo Choi 6.2_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | == Problem 3: An Uncommon PDF == | ||
| + | Let <math>Y</math> be a random variable with probability density function (PDF) | ||
| + | |||
| + | <math> | ||
| + | f_Y(v) = \left\{\begin{array}{ll} | ||
| + | 1 + v,& -1\leq v\leq0,\\ | ||
| + | v,& 0<v\leq1,\\ | ||
| + | 0,& \mbox{otherwise}. | ||
| + | \end{array}\right.</math> | ||
| + | |||
| + | Find | ||
| + | *(a) <math>P(|Y| < 1/2)</math> | ||
| + | *(b) <math>P(Y > 0|Y < 1/2)</math> | ||
| + | *(c) <math>E[Y]</math>. | ||
| + | |||
| + | *[[Anand Gautam 6.3_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | *[[Nicholas Browdues 6.3_ECE302Fall2008sanghavi]] | ||
| + | *[[Hamad AL Shehhi_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | *[[Ken Pesyna_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | *[[Kunal Kapoor 6.3_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | *[[Monsu Mathew 6.3_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | == Problem 4: Gaussian Coordinates == | ||
| + | A random point <math>(X,Y)</math> on a plane is chosen as follows: <math>X</math> and <math>Y</math> are chosen independently, with each one being a Gaussian random variable with zero mean and variance of 1. Let <math>D</math> be the square of the (random) distance of the point from the center. Find the PDF of <math>D</math>. Is <math>D</math> one of the common random variables? | ||
| + | |||
| + | *[[Katie Pekkarinen 6.4_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | *[[Divyanshu Kamboj 6.4_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | *[[Umang Jhunjhunwala 6.4_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | *[[Spencer Mitchell 6.4_ECE302Fall2008sanghavi]] | ||
| + | |||
| + | *[[Steven Streeter 6.4_ECE302Fall2008sanghavi]] | ||
| + | ---- | ||
| + | [[Main_Page_ECE302Fall2008sanghavi|Back to ECE302 Fall 2008 Prof. Sanghavi]] | ||
Latest revision as of 11:57, 22 November 2011
Contents
Instructions
Homework 6 can be downloaded here on the ECE 302 course website.
Problem 1: Ceiling of an Exponential
$ X $ is an exponential random variable with paramter $ \lambda $. $ Y = \mathrm{ceil}(X) $, where the ceiling function $ \mathrm{ceil}(\cdot) $ rounds its argument up to the closest integer, i.e.:
$ \mathrm{ceil}(a) $ = $ a $ if $ a $ is an integer = the smallest integer bigger than $ a $ if $ a $ is not an integer
What is the PMF of $ Y $? Is it one of the common random variables? (Hint: for all $ k $, find the quantity $ P(Y > k) $. Then find the PMF)
Problem 2: Fair Wages
``I do not have problems with anyone earning above average, as long as no one earns below average." - a quote (mistakenly attributed to) Max Weber. Can such a situation occur? Justify your answer.
- Brian Thomas 6.2_ECE302Fall2008sanghavi One possible solution
- Gregory Pajot 6.2_ECE302Fall2008sanghavi
- Virgil Hsieh 6.2_ECE302Fall2008sanghavi
- Zhongtian Wang 6.2_ECE302Fall2008sanghavi
- Michael Allen 6.2_ECE302Fall2008sanghavi
- Christopher Wacnik 6.2_ECE302Fall2008sanghavi
- Sahil Khosla 6.2_ECE302Fall2008sanghavi
- AJ Hartnett 6.2 --Different answer than above!_ECE302Fall2008sanghavi
- Jaewoo Choi 6.2_ECE302Fall2008sanghavi
Problem 3: An Uncommon PDF
Let $ Y $ be a random variable with probability density function (PDF)
$ f_Y(v) = \left\{\begin{array}{ll} 1 + v,& -1\leq v\leq0,\\ v,& 0<v\leq1,\\ 0,& \mbox{otherwise}. \end{array}\right. $
Find
- (a) $ P(|Y| < 1/2) $
- (b) $ P(Y > 0|Y < 1/2) $
- (c) $ E[Y] $.
Problem 4: Gaussian Coordinates
A random point $ (X,Y) $ on a plane is chosen as follows: $ X $ and $ Y $ are chosen independently, with each one being a Gaussian random variable with zero mean and variance of 1. Let $ D $ be the square of the (random) distance of the point from the center. Find the PDF of $ D $. Is $ D $ one of the common random variables?
