Computer Assignment $1$: The Gaussian Random Variable

Generation of random variables is of paramount importance in simulating communication systems. The

objective of this assignment is to visualize and generate a standard normal random variable, and plot its

probability density function $($PDF$)$ from the generated data. The following tasks will require the use of Matlab

to generate a standard Gaussian random variable $xN(0,1).$ Attach all figures with the corresponding

MATLAB code.

Task $1$: Use Matlab to generate a standard Gaussian random $x,$ i$.$e$.$ a sufficiently large vector of normally

distributed random variables. Use the generated data to plot the random variable $x.$

Task $2$: Use the generated data in Task $1$ to plot the PDF of $x.$

Hint: you may find these Matlab functions helpful for Task $1$ and Task $2,$ randn$(.)$ and histogram$(.).$

Task $3$: Plot the theoretical PDF of this random variable $-$ use suitable values for $x.$ Overlap the theoretical

plot with the simulated plot generated in Task $2.$ Recall the PDF of $x$ is

$f_x(x)=\frac{1}{\sqrt[\phantom{2}]{2}}{e}^{-\frac{x^2}{2}}$

Task $4$: Use Matlab to calculate the total area under the plotted curve in Task $3.$ This should represent the

total area $($or probability$)$ under the PDF$.$ Hint: you may find this function helpful for this step trapz$(...).$ For this task,

simply write down the output of this function.

Task $5($optional$)$: Use the Matlab function trapz$(...)$ along with the theoretical PDF plotted in Task $3$ to

calculate the probability $P(x>2).$ Use the Q$-$function to verify this result. For this task, simply compare

the two results. Are they the same?