## Random thoughts

September 04, 2011 at 06:23 PM | categories: statistics | View Comments

## Contents

- Random thoughts
- Random numbers in Matlab
- Uniformly distributed pseudorandom numbers
- Single random numbers
- Arrays of random numbers
- random integers uniformly distributed over a range
- Normally distributed numbers
- Comparing the random distribution to the normal distribution
- some properties of the normal distribution
- Summary

## Random thoughts

John Kitchin

clear all; close all; clc

## Random numbers in Matlab

Random numbers are used in a variety of simulation methods, most notably Monte Carlo simulations. In another later post, we will see how we can use random numbers for error propogation analysis. First, we discuss two types of pseudorandom numbers we can use in Matlab: uniformly distributed and normally distributed numbers.

## Uniformly distributed pseudorandom numbers

the :command:`rand` generates pseudorandom numbers uniformly distributed between 0 and 1, but not including either 0 or 1.

## Single random numbers

Say you are the gambling type, and bet your friend $5 the next random number will be greater than 0.49

n = rand if n > 0.49 'You win!' else 'You lose ;(' end

n = 0.6563 ans = You win!

## Arrays of random numbers

The odds of you winning the last bet are slightly stacked in your favor. There is only a 49% chance your friend wins, but a 51% chance that you win. Lets play the game 100,000 times and see how many times you win, and your friend wins. First, lets generate 100,000 numbers and look at the distribution with a histogram.

N = 100000; games = rand(N,1); nbins = 20; hist(games,nbins) % Note that each bin has about 5000 elements. Some have a few more, some a % few less.

Now, lets count the number of times you won.

wins = sum(games > 0.49) losses = sum(games <= 0.49) sprintf('percent of games won is %1.2f',(wins/N*100)) % the result is close to 51%. If you run this with 1,000,000 games, it gets % even closer.

wins = 51038 losses = 48962 ans = percent of games won is 51.04

## random integers uniformly distributed over a range

the rand command gives float numbers between 0 and 1. If you want a random integer, use the randi command.

randi([1 100]) % get a random integer between 1 and 100 randi([1 100],3,1) % column vector of three integers randi([1 100],1,3) % row vector of three integers

ans = 65 ans = 30 89 85 ans = 62 5 97

## Normally distributed numbers

The normal distribution is defined by where is the mean value, and is the standard deviation. In the standard distribution, and .

mu = 0; sigma=1;

get a single number

R = mu + sigma*randn

R = -0.4563

get a lot of normally distributed numbers

N = 100000; samples = mu + sigma*randn(N,1);

We define our own bins to do the histogram on

```
bins = linspace(-5,5);
% and get the counts in each bin also
counts = hist(samples, bins);
```

## Comparing the random distribution to the normal distribution

to compare our sampled distribution to the analytical normal distribution, we have to normalize the histogram. The probability density of any bin is defined by the number of counts for that bin divided by the total number of samples divided by the width

the bin width is the range of bins divided by the number of bins.

bin_width = (max(bins)-min(bins))/length(bins); plot(bins,1/sqrt(2*pi*sigma^2)*exp(-((bins-mu).^2)./(2*sigma^2)),... bins,counts/N/bin_width,'bo') legend('analytical distribution','Sampled distribution') % For the most part, you can see the two distributions agree. There is some % variation due to the sampled distribution having a finite number of % samples. As you increase the sample size, the agreement gets better and % better.

## some properties of the normal distribution

What fraction of points lie between plus and minus one standard deviation of the mean?

samples >= mu-sigma will return a vector of ones where the inequality is true, and zeros where it is not. (samples >= mu-sigma) & (samples <= mu+sigma) will return a vector of ones where there is a one in both vectors, and a zero where there is not. In other words, a vector where both inequalities are true. Finally, we can sum the vector to get the number of elements where the two inequalities are true, and finally normalize by the total number of samples to get the fraction of samples that are greater than -sigma and less than sigma.

```
a1 = sum((samples >= mu-sigma) & (samples <= mu+sigma))/N*100;
sprintf('%1.2f %% of the samples are within plus or minus one standard deviation of the mean.',a1)
```

ans = 68.44 % of the samples are within plus or minus one standard deviation of the mean.

What fraction of points lie between plus and minus two standard deviations of the mean?

a2 = sum((samples >= mu - 2*sigma) & (samples <= mu + 2*sigma))/N*100; sprintf('%1.2f %% of the samples are within plus or minus two standard deviations of the mean.',a2) % this is where the rule of thumb that a 95% confidence interval is plus or % minus two standard deviations from the mean.

ans = 95.48 % of the samples are within plus or minus two standard deviations of the mean.

## Summary

Matlab has a variety of tools for generating pseudorandom numbers of different types. Keep in mind the numbers are only pseudorandom, but they are still useful for many things as we will see in future posts.

% categories: statistics % tags: math