MA590 Homework 8 Solved

Description

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   (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(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.