This document contains 6 problems related to probability density functions, covariance matrices, random processes, and power spectral densities. Problem 1 involves finding expressions for the expected value and covariance of jointly Gaussian random variables X and Y with a given joint PDF. Problem 2 examines whether random variables X and Y with another given joint PDF are independent, uncorrelated, or orthogonal. Problem 3 involves determining whether a given matrix is a valid covariance matrix and finding properties of Gaussian random vectors. The remaining problems involve properties of random processes such as wide-sense stationarity, power spectral densities, and innovation representations.

1. Assignment 1
1. Suppose andX Y are jointly Gaussian random variables with the joint probability
density function given by
2 2( ) ( )( ) ( )
1 1 1 2 2
2 2 22(1 ) 1 21 2
2
1 2
2
1
, 2 1
( , ) .
x x y y
X Yf x y e
   
   

  
   

 
   
  


Find the expression for
(a) ( )E XY and cov( , )X Y
(b) / ( / )Y Xf y x
(c) ( / )E Y X x
2. Suppose andX Y are jointly random variables with the joint probability density
function given by
2 212 2
4
, 4( , ) .
x yx y
X Yf x y e
    

Examine if X and Y are independent, uncorrelated and orthogonal.
.
3. (a) Examine if
2 1 1
1 2 1
1 1 2
 
 
 
  
is a covariance matrix.
(b) Suppose X is a Gaussian random vector with
2
3
E
 
  
 
X and
2 1
1 3
 
  
 
XC .
(i) Find the linear transformation
1
2
Z
Z
 
 
 
= PX such that 1Z and 2Z are
uncorrelated.
(ii) Find 1 2, 1 2( , ).Z Zf z z
(d) Suppose ~ [0, 1]X U and 5
Y X is a function of X . Examine if X and Y
are uncorrelated.
4. A random process  ( )X t is defined in by
0 0( ) cos sinX t A t B t  
where 0 is a constant and A and B are zero-mean and jointly Gaussian random
variables with variances 2
A and 2
B .
(a) Examine if  ( )X t is WSS.
(b) Repeat part if 2 2
A B  .

2. 5. (a) A zero-mean WSS process { ( )}X t with the autocorrelation function
|
( ) 2XR e

 
 is input to a linear time invariant system with the frequency
response
1
( )
2
H j
j




. If { ( )}Y t is the output process, find
(i) the input PSD ( )XS 
(ii) the output PSD ( )YS 
(iii) the mean Y and the autocorrelation function ( )YR  .
6. (a) Write down the expression for the PSD and the autocorrelation function of a
continuous-time white noise process. Why is this process not realizable?
(b) A discrete-time white Gaussian noise process { }nX has the PSD ( )XS 
given by
10
( ) / ,
2
XS watt radian   

   
Roughly sketch the PSD ( )XS  , the autocorrelation function ( )XR  and the
first-order PDF ( )nXf x . Find the average power of the process.
(c) Suppose the generalized power spectral density ( )XS z of a discrete-time
random process { [ ]}X n is given by
1
( ) 2.5 , 0XS z z z z
    .
Use the spectral factorization theorem to have an innovation representation of
the process.