Understanding the Relationship Between PDF and CDF At its core, the CDF is just the accumulated area under the PDF . Definitions Let X be a continuous random variable. PDF: f X (x) CDF: F X (x) = P(X ≤ x) Mathematical Relationship The CDF is the integral of the PDF: F X (x) = ∫ -∞ x f X (t) dt Step-by-Step Method Identify the support of the PDF Determine where f X (x) is nonzero. Integrate the PDF This often gives: F X (x) = 0, x < a ∫ a x f X (t) dt, a ≤ x ≤ b 1, x > b Check endpoints F X (-∞) = 0 F X (∞) = 1 Example Giv...

How to Find E[X] and E[X²] This guide explains how to compute Expectation of X and Expectation of X² for practical GATE exam problems. Step 1: Identify What’s Given In practical problems, the random variable X is usually: Discrete (PMF or table) Continuous (PDF) A function of another random variable Case 1: Discrete Random Variable E[X] = Σ x i p i E[X²] = Σ x i ² p i Example X -1 0 2 P(X) 0.2 0.3 0.5 E[X] = (-1)(0.2) + 0(0.3) + 2(0.5) = 0.8 E[X²] = (1)(0.2) + 0 + (4)(0.5) = 2.2 Case 2: Continuous Random Variable E[X] = ∫ x f(x) dx E[X²] = ∫ x² f(x)...

Difference Between Distribution Function and Probability Distribution Function 1. Distribution Function (CDF) Definition: The distribution function (also called the Cumulative Distribution Function, CDF) is defined as: F X (x) = P(X ≤ x) It gives the probability that a random variable takes a value less than or equal to x . Properties: Non-decreasing function 0 ≤ F X (x) ≤ 1 F X (−∞) = 0 F X (+∞) = 1 2. Probability Distribution Function The probability distribution function describes how probability is distributed over values of a random variable. It depends on whether the random variable is discrete or continuous . (a) Discrete Random Variable — PMF Definition: P(X = x i ) = p(x i ) Example: X = {0, 1, 2} P(X = 0) = 0.2,  P(X = 1) = 0.5,  P(X...

Continuous Random Variable Mean (Expectation): E[X] = ∫ a b x f(x) dx Second Moment: E[X 2 ] = ∫ a b x 2  f(x) dx Variance: Var(X) = E[X 2 ] − (E[X]) 2 Mean & Variance Calculator (Continuous PDF) Enter the PDF f(x) using JavaScript syntax (use x ). Lower limit (a): Upper limit (b): PDF f(x): Calculate Discrete Random Variable Mean (Expectation): E[X] = ∑ x i p i Second Moment: E[X 2 ] = ∑ x i 2 p i Variance: Var(X) = E[X 2 ] − (E[X]) 2 Mean & Variance Calculator (Discrete PMF) Enter values separated by commas. Values of X: -1, 0, 2 Probabilities P(X): 0.2, 0.3, 0.5 Calculate

  MATLAB Code clc; clear all; close all; % Number of samples to generate n = 100000; % Generate Uniform distribution between 0 and 1 r = rand(1, n);  % rand generates numbers in the range [0, 1] % Transform to the range [-1, 1] a = -1; b = 1; uniform_values = a + (b - a) * r; % Plot the histogram of the generated uniform distribution figure; histogram(uniform_values, 30, 'Normalization', 'pdf');  % Normalized to show probability density title('Uniform Distribution between -1 and 1'); xlabel('Value'); ylabel('Probability Density'); % Generate Gaussian distribution (Standard Normal Distribution) gaussian_values = randn(1, n);  % Standard normal distribution (mean = 0, std = 1) % 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...

📘 Overview 📘 Mean, Variance 📘 SNR Definition, and Noise Variance Calculation, and SNR dB to Linear 📘 SNR dB to Linear 📘 MATLAB Code 📚 Further Reading What is Gaussian Noise? Gaussian noise is a random signal whose amplitude follows a Gaussian (normal) distribution . p(x) = (1 / √(2πσ²)) e -(x-μ)² / (2σ²) Where: μ = mean σ² = variance It is widely used in communication systems because many natural noise sources follow this distribution. Difference Between Gaussian Noise and AWGN Feature Gaussian Noise AWGN Definition Noise with Gaussian distribution Gaussian + Additive + White Additive Not necessarily Always additive White (flat spectrum) Not required Yes Usage General noise model Communication systems AWGN Noise: Mean and Variance in Practical Systems In practical communication systems, Additive White Gaussian Noise (AWGN) is modeled with a zero mean and varia...

  Box-Mullar Transform generates random noise where noise samples are independent, standard, and normally distributed. So, we can say this type of noise is Gaussian noise.   In MATLAB Codes below, independent random variables u1 and u2 are uniformly distributed between 0 and 1 and then transformed using the Box-Muller method to obtain pairs of independent, standard, normally distributed random variables z0 and z1.   MATLAB Code for Generating Random (Gaussian) Noise clc; clear all; close all; numSamples = 1000; % Number of samples gaussianNoise = generateGaussianNoise(numSamples); disp(gaussianNoise); function noise = generateGaussianNoise(numSamples) % Generate Gaussian noise using Box-Muller transform noise = zeros(1, numSamples); % Generate pairs of independent, standard, normally distributed random variables for i = 1:2:numSamples u1 = rand(); u2 = rand(); z0 = sqrt(-2 * log(u1)) * cos(2 * pi * u2); z1 = sqrt(-2 * log(u1)) * sin(2 * pi * u2); noise(i) ...

📘 Introduction, Gaussian Distribution, Relationship Between Gaussian and Rayleigh Distribution 🧮 How to mitigate Rayleigh fading? 🧮 Equalizer to reduce Rayleigh Fading (or Multi-path Effects) in MATLAB 🧮 MATLAB Code for Effects of AWGN and Rayleigh Fading in Wireless Communication 🧮 Simulator for the effect of AWGN and Rayleigh Fading on a BPSK Signal 📚 Further Reading Wireless Signal Processing Gaussian and Rayleigh distributions ...   The Rayleigh distribution in classical fading models (like wireless communication) arises from modeling the real and imaginary parts of a complex baseband signal as independent, zero-mean Gaussian random variables — under specific assumptions . 1. Gaussian Distribution  The Gaussian distribution has a lot of applications in wireless communication. Since noise in wireless communication systems is unpredictable, we frequently assume that it has a Gaussian distribution...

📘 Introduction, AWGN, and Rayleigh Fading 🧮 Simulator for the effect of AWGN and Rayleigh Fading on a BPSK Signal 🧮 MATLAB Codes 📚 Further Reading Wireless Signal Processing Gaussian and Rayleigh Distribution Difference between AWGN and Rayleigh Fading 1. Introduction Rayleigh fading coefficients and AWGN, or Additive White Gaussian Noise (AWGN) in Wireless Channels , are two distinct factors that affect a wireless communication channel. In mathematics, we can express it in that way. Fig: Rayleigh Fading due to multi-paths Let's explore wireless communication under two common noise scenarios: AWGN (Additive White Gaussian Noise) and Rayleigh fading. y = h*x + n ... (i) Symbol '*' represents convolution. The transmitted signal x is multiplied by the channel coeffic...

Wireless Signal Processing Gaussian and Rayleigh Distribution MATLAB Code for Gaussian Random Variable ...   We are all aware that a noise signal is added to a signal as it is transmitted from transmitter to receiver, especially when using a wireless channel. Although we can't entirely eliminate such noise signals. With a better understanding of noise, its pattern, etc., we may be able to recover the original transmitted data. For a typical wireless communication process,  y = x + n where x and y are the transmitted and received signals, respectively, and n stands for noise.   We can see that the standard deviation and mean of the gaussian noise represent the entirety of the noise pattern in the abovementioned PDF of the gaussian random variable. These two variables are crucial for almost all noise types. The sample values' standard deviation indicates how they vary from one another. It offers us a general sense of the range in which...

Home / Wireless Communication  / Gaussian random variable and its PDF Wireless Signal Processing Gaussian and Rayleigh Distribution 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...