Uniform random variable - GaussianWaves (2024)

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Uniform random variables are used to model scenarios where the expected outcomes are equi-probable. For example, in a communication system design, the set of all possible source symbols are considered equally probable and therefore modeled as a uniform random variable.

The uniform distribution is the underlying distribution for an uniform random variable. A continuous uniform random variable, denoted as , take continuous values within a given interval , with equal probability. Therefore, the PDF of such a random variable is a constant over the given interval is.

$$ f_X(x) = \begin{cases}\frac{1}{b-a} & \text{when } a < x < b\\0 & \text{otherwise} \end{cases} $$

In Matlab, rand function generates continuous uniform random numbers in the interval . The rand function picks a random number in the interval in which the probability of occurrence of all the numbers in the interval are equally likely. The command rand(n,m) will generate a matrix of size . To generate a random number in the interval one can use the following expression.

a + (b-a)*rand(n,m); %Here nxm is the size of the output matrix

To test whether the numbers generated by the continuous uniform distribution are uniform in the interval , one has to generate very large number of values using the rand function and then plot the histogram. The Figure 1 shows that the simulated PDF and theoretical PDF are in agreement with each other.

a=2;b=10; %open interval (2,10)X=a+(b-a)*rand(1,1000000);%simulate uniform RV[p,edges]=histcounts(X,'Normalization','pdf');%estimated PDFoutcomes = 0.5*(edges(1:end-1) + edges(2:end));%possible outcomesg=1/(b-a)*ones(1,length(outcomes)); %Theoretical PDFbar(outcomes,p);hold on;plot(outcomes,g,'r-');title('Probability Density Function');legend('simulated','theory');xlabel('Possible outcomes');ylabel('Probability of outcomes');

On the other hand, a discrete random variable generates discrete values that are equally probable. The underlying discrete uniform distribution is denoted as , where , is a finite set of discrete elements that are equally probable as described by the probability mass function (PMF)

$$f_X(x)= \begin{cases}\frac{1}{n} & \text{where } x \in {s_1,s_2,…,s_n } \\ 0 & otherwise \end{cases} $$

There exist several methods to generate discrete uniform random numbers and two of them are discussed here. The straightforward method is to use randi function in Matlab that can generate discrete uniform numbers in the integer set . The second method is to use rand function and ceil the result to discrete values. For example, the command to generate uniformly distributed discrete numbers from the set is

X=ceil(n*rand(1,100));

The uniformity test for discrete uniform random numbers can be performed and it is very similar to the code shown for the continuous uniform random variable case. The only difference here is the normalization term. The histogram values should not be normalized by the total area under the histogram curve as in the case of continuous random variables. Rather, the histogram should be normalized by the total number of occurrences in all the bins. We cannot normalized based on the area under the curve, since the bin values are not dense enough (bins are far from each other) for proper calculation of total area. The code snippet is given next. The resulting plot (Figure 2) shows a good match between the simulated and theoretical PMFs.

X=randi(6,100000,1); %Simulate throws of dice,S={1,2,3,4,5,6}[pmf,edges]=histcounts(X,'Normalization','pdf');%estimated PMFoutcomes = 0.5*(edges(1:end-1) + edges(2:end));%S={1,2,3,4,5,6}g=1/6*ones(1,6); %Theoretical PMFbar(outcomes,pmf);hold on;stem(outcomes,g,'r-');title('Probability Mass Function');legend('simulated','theory');xlabel('Possible outcomes');ylabel('Probability of outcomes');

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Topics in this chapter

Random Variables - Simulating Probabilistic Systems
● Introduction
● Plotting the estimated PDF
● Univariate random variables
□ Uniform random variable
□ Bernoulli random variable
□ Binomial random variable
□ Exponential random variable
□ Poisson process
□ Gaussian random variable
□ Chi-squared random variable
□ Non-central Chi-Squared random variable
□ Chi distributed random variable
□ Rayleigh random variable
□ Ricean random variable
□ Nakagami-m distributed random variable
● Central limit theorem - a demonstration
● Generating correlated random variables
□ Generating two sequences of correlated random variables
□ Generating multiple sequences of correlated random variables using Cholesky decomposition
● Generating correlated Gaussian sequences
□ Spectral factorization method
□ Auto-Regressive (AR) model

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