$2\mathrm{.}7$ The central limit theorem states that if random variables $x_i,1in,$ are i$.$i$.$d$.$

$($independent and identically distributed$),$ with finite mean and variance, and $n$ is large,

then their average $($i$.$e$.,Y=\frac{1}{n}{\displaystyle \underset{i=1}{\overset{n}{}}}{x}_i)$ has roughly a Gaussian distribution. This

theorem explains why thermal noise generated in electric circuits has a Gaussian dis$-$

tribution. In this problem we verify this theorem using MATLAB.

a$.$ Using MATLAB, generate a vector $x$ of length $1,000,000$ whose components

are uniform random variables distributed between $0$ and $1.$ The components of

this vector are the $x_i\text{'}$s in the central limit theorem. $($Use MATLAB command

"rand" to generate the sequence.$)$

b$.$ Find the average of every $100$ consecutive elements of $x$ and generate the se$-$

quence $y$ of length $10,000.$ So$,Y_1$ is the average of $x_1$ through $x_{100},Y_2$ is the

average of $x_{101}$ through $x_{200},$ and so on$.$

I need to matlab code