Home / Wireless Communication / Gaussian random variable and its PDF
What exactly are Gaussian Random Variable and its probability distribution function (PDF) are
The practical communication system is modeled as
y = x + n;
Where y=received signal
x= transmitted signal
n= noise
What is the significance of the Gaussian Random Variable?
We know, especially for wireless communication, whenever we transmit a signal from transmitter to receiver, there will be some additive white Gaussian noise to the signal when we receive it from the receiver. The additive white Gaussian noise has some properties, like zero mean and a specific standard deviation. We learn later what exactly they mean, what Deviations are, and the relation of the Gaussian random variable with it. Here, the word "random" is used because noise is always unexpected in the communication system. We can't predict it before the transmission of the signal. But we can draw its probability distribution function (PDF) from several experiments or values.
What exactly is Gaussian Random Variable PDF is
PDF of Gaussian random variable is defined as
Here, σ = Standard Deviation of random variable samples
μ = mean of random variable samples
In the above figure probability distribution function of the Gaussian random variable is shown. Students often need clarification with the title of the x label and y label. x tag defines the variation of the standard deviation value of Gaussian noise collected from large samples or populations or many experiments. After getting the standard Deviation of noise,e we plot the probability of standard deviations derived from large samples.
MATLAB Code for gaussian random variable and its PDF
clear;
close all;
% Number of samples to generate
n = 100000;
% Generate Gaussian distribution (Standard Normal Distribution)
gaussian_values = randn(1, n); % Standard normal distribution (mean = 0, std = 1)
% Calculate mean and standard deviation of the Gaussian values
mu = mean(gaussian_values);
sigma = std(gaussian_values);
% Calculate the range for 1, 2, and 3 standard deviations
range_1sigma = sum(gaussian_values >= (mu - sigma) & gaussian_values <= (mu + sigma)) / n * 100; % Percentage within 1 standard deviation
range_2sigma = sum(gaussian_values >= (mu - 2*sigma) & gaussian_values <= (mu + 2*sigma)) / n * 100; % Percentage within 2 standard deviations
range_3sigma = sum(gaussian_values >= (mu - 3*sigma) & gaussian_values <= (mu + 3*sigma)) / n * 100; % Percentage within 3 standard deviations
% Display the results
fprintf('Percentage of values within 1 standard deviation: %.2f%%\n', range_1sigma);
fprintf('Percentage of values within 2 standard deviations: %.2f%%\n', range_2sigma);
fprintf('Percentage of values within 3 standard deviations: %.2f%%\n', range_3sigma);
% Plotting the Gaussian distribution
figure;
histogram(gaussian_values, 30, 'Normalization', 'pdf'); % Normalized to show probability density
title('Gaussian Distribution (Standard Normal)');
xlabel('Value');
ylabel('Probability Density');
Output
Percentage of values within 1 standard deviation: 68.36%
Percentage of values within 2 standard deviations: 95.40%
Percentage of values within 3 standard deviations: 99.73%
Copy the aforementioned MATLAB Code from here003
Real-world mathematical examples to understand mean and standard Deviation
Mean of a Random Variable
As we have mentioned above, noise is random in a communication system. So, we take hundreds of values of that parameter and draw a PDF. For example, we have received ten random variables, i.e., X1, X2, X3, X4,..., X9, and X10. Then we calculate its mean or average. That is also meaningful.
Xmean = (X1 + X2 + X3+... +X8 +X9 +X10)/10
Standard Deviation of a Random Variable
In electronic communication, the standard signal deviation tells us how the signal varies over time. For example, we measure a signal in different time instants, from a different position, or at another aspect. Then we can calculate the standard Deviation to see how the signal varies. That value also matters for electronic devices. Similarly, we calculate the standard deviation value from many samples in the case of a Gaussian random variable. For example, in a class, marks obtained in math by students are as follows:
Student 1: 92 out of 100
Student 2: 85 out of 100
Student 3: 74 out of 100
Student 4: 70 out of 100
Student 5: 60 out of 100
Student 6: 66 out of 100
Student 7: 82 out of 100
Student 8: 63 out of 100
Student 9: 76 out of 100
Student 10: 59 out of 100
The average marks obtained by students are calculated as
=(92+85+74+70+60+66+82+63+76+59)/10
=72.7
The mean value is 72.7
Now, we'll calculate Standard Deviation,
Std or σ= sqrt{(1/(N-1) * Σ(Ni -N0)^2}
Here, N= total number of sample
Ni denotes the instantaneous value of N
N0 denotes the mean of N
'sqrt' denotes 'square root' here
The standard Deviation for obtained marks by students is,
Std or σ =sqrt{1/(10-1) * Σ (Ni -72.7)^2}
(as here several samples or population is 10 & mean/avg. =72.7)
Or, σ = sqrt [1/9 * {(92-72.7)^2 + (85-72.7)^2 + (74-72.7)^2 + (70-72.7)^2 + (60-72.7)^2 + ... +(76-72.7)^2 + (59-72.7)^2}]
Or, σ = 10.57
Standard Deviation, in many cases defined as the notation σ (sigma). The standard Deviation (σ ) indicates how far a 'typical' observation deviates from the data's average or mean value, μ.
