Difference between revisions of "Homework6 ECE302Fall2008sanghavi" - Rhea

Revision as of 14:58, 12 October 2008

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

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 (random) distance of the point from the center. Find the PDF of $ D $. Is $ D $ one of the common random variables?