MA 590
Homework 8
Your code should be appropriately commented to receive full credit.
Problems
1 (20 points) 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 ].
(b) If X and Y are independent, then X and Y are uncorrelated.
(c) Var(sX) = s
2 Var(X) for some scalar s
(d) Var(X + Y ) = Var(X) + Var(Y ) + 2 Cov(X, Y )
2 (10 points) 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(σ
2
).
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.
2
