(→Problem 3: An Uncommon PDF) |
(→Problem 3: An Uncommon PDF) |
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== Problem 3: An Uncommon PDF == | == Problem 3: An Uncommon PDF == | ||
Let <math>Y</math> be a random variable with probability density function (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 | Find | ||
Revision as of 07:24, 8 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.
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\left(|Y| < \frac12\right) $
- (b) $ P\left(Y > 0|Y < \frac12\right) $
- (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?
