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Gaussian random variable and its PDF


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What exactly are Gaussian Random Variable and its 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, Ïƒ = 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. 

You see values like, -10, -8, -6, ...., 0, +6, +8, and +10 on the x-axis. If you notice t, you can see that the probability of a standard Deviation of value 2 is around 0.18. Here actually, the likelihood of two times of standard deviation is 0.18. Similarly, the 2σ or 2 times the expected deviation probability is around 0.02.


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 Ïƒ= 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 Ïƒ =sqrt{1/(10-1) * Î£ (Ni -72.7)^2} 

(as here several samples or population is 10 & mean/avg. =72.7)

Or, Ïƒ = 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, Ïƒ = 10.57

Standard Deviation, in many cases defined as the notation Ïƒ (sigma). The standard Deviation (σ ) indicates how far a 'typical' observation deviates from the data's average or mean value, Î¼.



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