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Introduction :: Exercises

Introduction

This exercise will provide an opportunity to do some calculations and plots with actual data. The intent is to make some of the abstract concepts a little more concrete. This exercise is to be done using MATLAB.

Background

randn

Every call to the MATLAB function rand generates an independent instance of a standard Gaussian random variable. That is, randn is a Gaussian random number generator: $X \sim \Nc(0,1)$ . Random column vectors of length n are generated by randn(n,1). Row vectors are generated with randn(1,n). A matrix of random numbers is generated with randn(n,m). For more information, type help randn in MATLAB.

rand

Similarly, the MATLAB function rand generates independent uniform $\Uc(0,1)$ random numbers. Column and row vectors and matrices of random numbers are generated using rand(n,1), rand(1,n), rand(n,m). For more information type help rand in MATLAB.

Histograms

The MATLAB hist command produces a histogram. This is a representation of the empirical density function. In a histogram, a sequence of bins is established. For each value in a set of data, the number of times that data points fall in a bin is counted. In the MATLAB hist command, the histogram is plotted automatically. To see an example of how the histogram works, type the following in MATLAB:

x = randn(1,1000);   % create a vector of 1000 Gaussian random numbers
hist(x,20);  % plot the histogram with 20 bins
hist(x,100); % plot the histogram with 100 bins

Estimating mean and covariance

Given a sequence of vector observations $\xbf_1, \xbf_2, \ldots, \xbf_N$ , where each vector is a column vector of length

drawn independently and identically distributed according to some distribution, the sample mean of the distribution is

$\displaystyle \hat{\mubf} = \frac{1}{N} \sum_{i=1}^N \xbf_i.$

The sample covariance is

$\displaystyle \hat{\Sigma} = \frac{1}{N-1} \sum_{i=1}^N (\xbf_i - \mubfhat)(\xbf_i - \mubfhat)^T.$

Copyright 2008, Todd Moon. Cite/attribute Resource. admin. (2006, June 13). Programming Assignments. Retrieved November 24, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/program1_1.htm. This work is licensed under a Creative Commons License.

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	Info Programming Assignments Document Actions Introduction :: Exercises Introduction This exercise will provide an opportunity to do some calculations and plots with actual data. The intent is to make some of the abstract concepts a little more concrete. This exercise is to be done using MATLAB. Background `randn` Every call to the MATLAB function `rand` generates an independent instance of a standard Gaussian random variable. That is, `randn` is a Gaussian random number generator: $X \sim \Nc(0,1)$ . Random column vectors of length `n` are generated by `randn(n,1)`. Row vectors are generated with `randn(1,n)`. A matrix of random numbers is generated with `randn(n,m)`. For more information, type `help randn` in MATLAB. `rand` Similarly, the MATLAB function `rand` generates independent uniform $\Uc(0,1)$ random numbers. Column and row vectors and matrices of random numbers are generated using `rand(n,1)`, `rand(1,n)`, `rand(n,m)`. For more information type `help rand` in MATLAB. Histograms The MATLAB `hist` command produces a histogram. This is a representation of the empirical density function. In a histogram, a sequence of bins is established. For each value in a set of data, the number of times that data points fall in a bin is counted. In the MATLAB `hist` command, the histogram is plotted automatically. To see an example of how the histogram works, type the following in MATLAB: x = randn(1,1000); % create a vector of 1000 Gaussian random numbers hist(x,20); % plot the histogram with 20 bins hist(x,100); % plot the histogram with 100 bins Estimating mean and covariance Given a sequence of vector observations $\xbf_1, \xbf_2, \ldots, \xbf_N$ , where each vector is a column vector of length drawn independently and identically distributed according to some distribution, the sample mean of the distribution is $\displaystyle \hat{\mubf} = \frac{1}{N} \sum_{i=1}^N \xbf_i.$ The sample covariance is $\displaystyle \hat{\Sigma} = \frac{1}{N-1} \sum_{i=1}^N (\xbf_i - \mubfhat)(\xbf_i - \mubfhat)^T.$ Copyright 2008, Todd Moon. Cite/attribute Resource. admin. (2006, June 13). Programming Assignments. Retrieved November 24, 2009, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Stochastic_Processes/program1_1.htm. This work is licensed under a Creative Commons License.	Reuse Course Download this course

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Introduction

Background