Generate and plot the following:

1. A histogram for Gaussian "Normal" random variable X_1 with zero-mean and unit variance. Calculate its mean and variance.
2. Repeat 1 for a uniformly distributed random variable X_2 on the interval [-1,1].
3. Repeat 1 for an exponentially distributed random variable X_3 with a = 100 .

Apply the following transformations on X_1 ::
Y_1=2X_1+3
Y_2=X_1^2

Show the histograms of Y_1 and Y_2 . Calculate their means and variances and comment on your findings.

Verify the central limit theorem by performing the following tasks:
Generate 100 Gaussian "Normal" random variables (where each is of zero-mean and unit variance) and add them. Show the histograms for the first and the summed random variables. What is the mean and variance of the resultant random variable?
Repeat C(a) but with a sum of 2, 10, 100 random variables that are exponentially distributed with a = 1 . What is the mean and variance of the resultant random variables, i.e., for the three cases 2, 10 and 100?

You can use the following MATLAB functions:

random, hist, mean, var, the dot operator ".", the power operator "^", & for.

What to do?

Generate and plot PDFs for various random variables in MATLAB.

Verify the central limit theorem .

The results including all required figures.

For every figure write the results and observations .