Question

The Gaussian distribution also known as the Normal distribution, is given by the following

equation:

𝑦(𝑥) = 𝑒𝑥𝑝 −(𝑥−𝜇)^2/2𝜎^2

where parameter 𝝁 is the mean and 𝝈 the standard deviation.

(i) Write a MATLAB code to create a 1000 point Gaussian distribution of random numbers

having 𝜇 = 0 and 𝜎 = 1. (20)

(ii) Plot this distribution. (10)

(iii) Prove that the full width–half maximum (FWHM), of the above distribution is given by :

FWHM = 2𝜎√2ln 2 (10)

Answer 1

i: >>x=normrnd(0,1,1000);

ii:

>> y=@(x)exp(-x.^2/2);

>> x=-5:0.01:5;

>> plot(x,y(x))

$iii:\\\max \left( f\left( x \right) \right) =\max \left( \frac{1}{\sqrt{2\pi \sigma ^2}}\exp \left( -\frac{x^2}{2\sigma ^2} \right) \right) =f\left( 0 \right) =\frac{1}{\sqrt{2\pi \sigma ^2}}\\f\left( x \right) =\frac{1}{2}\max \left( f \right) \Rightarrow f\left( x \right) =\frac{1}{2\sqrt{2\pi \sigma ^2}}\Rightarrow \frac{1}{\sqrt{2\pi \sigma ^2}}\exp \left( -\frac{x^2}{2\sigma ^2} \right) =\frac{1}{2\sqrt{2\pi \sigma ^2}}\Rightarrow \\\Rightarrow \frac{x^2}{2\sigma ^2}=\ln 2\Rightarrow x=\pm \sigma \sqrt{2\ln 2}\Rightarrow FWHM=x_2-x_1=\sigma \sqrt{2\ln 2}-\left( -\sigma \sqrt{2\ln 2} \right) =2\sigma \sqrt{2\ln 2}$