Random Signals for Engineers using MATLAB and Mathcad
Copyright  1999 Springer Verlag NY
Example 3.8 Random Number Sequences with Arbitrary Distributions
In this example we will use the histogram function to create a discrete random number sequence
with and arbitrary distribution. Let us assume that there exists a probability outcome pair that we would
like to match
val=[1 2 3 4]
Probs=[.2 .1 .45 .25]
val =
1
2
Probs =
0.2000
3
0.1000
4
0.4500
0.2500
We set up a vector of intervals that are equal to the probabilities we would like to match. The cumsum is a
Matlab implementation of Equation 3.5-6 and the center points are chosen to use the histogram function to
determine the number of times random variable falls in each bin.
function y= center_bin(intv)
% computes the center bin for a histogram
n=length(intv)-1;
d= (intv(2)-intv(1))/2;
y(1)= intv(1)-d
for i=1:n
y(i+1) = intv(i)+d
d=intv(i+1)-y(i+1)
end
intervals=cumsum(Probs)
x=center_bin(intervals)
intervals =
0.2000
x =
0.1500
0.3000
0.7500
1.0000
0.2500
0.3500
1.1500
Now draw random numbers uniformly distributed between 0 and 1. If a number falls in the first interval,
interpret it as outcome 1, in the second interval as outcome 2, and so on. r is a random number sequence
from a uniform distribution
r=[0 .19 .59 .35 .82 .17 .71 .3]
r =
Columns 1 through 7
0
0.1900
Column 8
0.3000
0.5900
0.3500
0.8200
0.1700
0.7100
The histogram function counts the number of Ri in each interval 0-0.2, 0.2-0.3, 0.3-0.75 and 0.75-1. R is a
column vector
for i=1:length(r)
n=hist(r(i),x);
m(i)=val*n';
end
The histogram function forms a vector that represents the interval that the first column number of r falls.
The dot product by column of m and the val vector assigns the proper outcome to the r i as 1 2 3 4. The
vector m now contains the results of N trials of the experiment.
m
m =
1
1
3
3
4
1
3
2
Now let us repeat the experiment with a larger sample and indeed verify by the use of the counting function
that the sample follows the original distribution.
N=500;
r=rand(1,N);
for i=1:length(r)
n=hist(r(i),x);
m(i)=val*n';
end
We use the histogram function to count the number of 1’s, 2’s etc. in the m vector. Using the vector
operator of Matlab we divide each count by N to determine the relative frequency below and compare this
to the original probability definition.
v=hist(m,val)./N
v =
0.1800
0.0900
0.4940
0.2360
0.4500
0.2500
Probs = [.2 .1 .45 .25]
Probs =
0.2000
0.1000
For N = 500 there is good agreement as we increase N we expect the agreement to improve.
