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== Problem 4: Gaussian Coordinates == | == 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 (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? | + | 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]] | *[[Katie Pekkarinen 6.4_ECE302Fall2008sanghavi]] | ||
Revision as of 05:06, 15 October 2008
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
- Michael Allen 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?
