Your assignment should be wellorganized, typed (or neatly written and scanned) and saved as a .pdf for submission on Canvas. You must show all of your work to receive full credit. For problems requiring the use of MATLAB code, remember to also submit your .m-files on Canvas as a part of your completed assignment. Your code should be appropriately commented to receive full credit.
Problems
1 ( Given two random variables X and Y , prove the following. (a) The covariance of X and Y can be equivalently written as
Cov(X,Y ) = E[(X − x¯)(Y − y¯)]
or as
Cov(X,Y ) = E[XY ] − E[X]E[Y ]
where ¯x = E[X] and ¯y = E[Y ].
If X and Y are independent, then X and Y are uncorrelated.
Var(sX) = s2 Var(X) for some scalar s
Var(X + Y ) = Var(X) + Var(Y ) + 2 Cov(X,Y )
2 Consider a vector-valued random variable
A = Xe1 + Y e2
where e1 and e2 are the orthogonal Cartesian unit vectors, and X and Y are real-valued random variables with
X,Y ∼ N(0,σ2).
The random variable
R = ||A||2
is then distributed according to the Rayleigh distribution,
R ∼ Rayleigh(σ2).
Derive the analytic expression of the Rayleigh distribution, and write a MATLAB program that generates points from the Rayleigh distribution. Make a plot of the distribution and a histogram of the points you generated.
Note: For any of the above problems for which you use MATLAB to help you solve, you must submit your code/.m-files as part of your work. Your code must run in order to receive full credit. If you include any plots, make sure that each has a title, axis labels, and readable font size, and include the final version of your plots as well as the code used to generate them.
