EE 278B
Statistical Signal Processing
Tuesday, October 11, 2011
Handout #3
Homework #1 Solutions
1. (10 points) Exponential Random Variable.
a. It is easy to see that the CDF of X Exp() is FX (x) = 1 ex for x 0. For = 0.1,
we have Pcfw_X > 10 = 1 FX (10)
EE 278B
Statistical Signal Processing
October 20, 2011
Handout #6
Homework #4
Due Thursday, October 27
1. Coloring and whitening. Let
210
= 1 2 1 .
012
a. Find the coloring and whitening matrices of using the eigenvalue method discussed in
lecture slides
EE 278B
Statistical Signal Processing
October 18, 2011
Handout #5
Homework #2 Solutions
1. (5 points) First available teller. The tellers service times are exponentially distributed, hence
memoryless. Thus the service time distribution does not depend on
EE 278
Statistical Signal Processing
October 9, 2009
Handout #5
Homework #2 Solutions
1. The cdf of random variable X is given by
FX (x) =
1
3
2
+ 3 (x + 1)2
1 x 0
x < 1
0
a. Find the probabilities of the following events.
1
A = cfw_X > 3 ,
B = cfw_|X |
EE 278 Statistical Signal Processing Homework #8 Due: Wednesday, December 2
November 18, 2009 Handout #18
1. Discrete-time Wiener process. Let cfw_Zn : n 0 be a discrete-time white Gaussian noise process; that is, Z1 , Z2 , Z3 , . . . are i.i.d. N (0, 1).
EE 278
Statistical Signal Processing
October 24, 2015
Handout #9
Homework #4 Solutions
1. (10 points) Coloring and whitening.
a. We denote the eigenvalue and eigenvector matrices of as and U, respectively. After
using linear algebra methods (or Matlab, wh
EE 278B
Statistical Signal Processing
October 13, 2011
Handout #4
Homework #3
Due Thursday, October 20
1. Estimation vs. detection. Signal X and noise Z are independent random variables, where
X=
+1 with probability
1 with probability
1
2
1
,
2
and Z U[2,
EE 278 Statistical Signal Processing Homework #7 Solutions
November 20, 2009 Handout #19
1. Convergence examples. Consider the following sequences of random variables dened on the probability space (, F , P), where = cfw_0, 1, . . . , m 1, F is the collec
EE 278B
Statistical Signal Processing
October 25, 2011
Handout #7
Homework #3 Solutions
1. (15 points) Estimation vs. detection.
a. We can easily nd the piecewise constant density of Y
1 |y | 1
4
1
fY (y ) = 8 1 < |y | 3
0 otherwise
The conditional proba
EE 278B
Statistical Signal Processing
October 29, 2011
Handout #9
Homework #4 Solutions
1. (10 points) Coloring and whitening.
a. We denote the eigenvalue and eigenvector matrices of as and U , respectively. After
using linear algebra methods (or Matlab,
EE 278B
Statistical Signal Processing
September 29, 2011
Handout #1
Homework #1
Due Thursday, October 6
You can either hand the assignment to me after class or drop it in the
Homework In box in the EE 278B drawer of the class le cabinet on the
2nd oor of
EE 278 Statistical Signal Processing Homework #6 Due: Wednesday November 4
October 28, 2009 Handout #10
1. Gaussian random vector Suppose X N (, ) is a Gaussian random vector with 1 110 = 5 and = 1 4 0 . 2 009 a. Find the pdf of X1 . b. Find the pdf of X2
EE 278B
Statistical Signal Processing
Thursday, November 17, 2011
Handout #16
Homework #7
Due Thursday, December 1
1. Autocorrelation functions. Find the autocorrelation functions of
a. the process X (t) = At + B of problem 2 in homework 6.
b. the process
EE 278 Statistical Signal Processing Homework #1 Solutions
October 2, 2009 Handout #3
1. Monty Hall. (Bonus) Gold is placed behind one of three curtains. A contestant chooses one of the curtains. Monty Hall, the game host, opens an unselected empty curtai
EE 278B
Statistical Signal Processing
October 6, 2011
Handout #2
Homework #2
Due Thursday, October 13
1. First available teller. A bank has two tellers. The service times for tellers 1 and 2 are independent exponential random variables X1 Exp(1 ) and X2 E
EE 278 Statistical Signal Processing Homework #5 Solutions
October 30, 2009 Handout #12
1. Additive-noise channel with path gain. Consider the additive noise channel shown in the gure below, where X and Z are zero mean and uncorrelated, and a and b are co
EE 278B
Statistical Signal Processing
Thursday, November 10, 2011
Handout #14
Homework #6
Due Thursday, November 17
1. Vector CLT. The signal received over a wireless communication channel can be represented by
two sums
n
n
1
1
X1n =
Zj cos j and X2n =
EE 278B
Statistical Signal Processing
Friday, November 25, 2011
Handout #17
Homework #6 Solutions
1. (10 points) Vector CLT. The key point to this problem is to realize that we are asked to nd
the distribution of the random vector Yn = [ X1n X2n ]T as n .
EE 278B
Statistical Signal Processing
Tuesday, December 6, 2011
Handout #19
Homework #7 Solutions
1. (20 points) Autocorrelation functions.
a. The mean function is
X (t) = E[At + B ] = E[A]t + E[B ] = 0.
The autocorrelation function is
RX (t1 , t2 ) = E[(
Lecture Notes 3
Random Vectors
Specifying a Random Vector
Mean and Covariance Matrix
Coloring and Whitening
Gaussian Random Vectors
EE 278B: Random Vectors
31
Specifying a Random Vector
Let X1, X2, . . . , Xn be random variables defined on the same proba
EE 278
Statistical Signal Processing
November 3, 2007
Handout #11
Homework #5 Solutions
1. (10 points) Additive-noise channel with path gain. First we find the mean and variance of Y
2
and its covariance with X . In the following we use the notation X
= P
EE 278
Statistical Signal Processing
Tuesday, November 20, 2007
Handout #17
Homework #6 Solutions
1. (40 points) Gaussian random vector
a. The marginal pdfs of a jointly Gaussian pdf are Gaussian. Therefore X1 N (1, 1).
b. Since X2 and X3 are independent
EE 278
Statistical Signal Processing
October 4, 2007
Handout #2
Homework #2
Due Thursday, October 11
1. The cdf of random variable X is given by
(1 2
+ 3 (x + 1)2
3
FX (x) =
0
1 x 0
x < 1
a. Find the probabilities of the events
A = cfw_X > 31 ,
B = cfw_|
EE 278
Statistical Signal Processing
October 25, 2007
Handout #8
Homework #5
Due Thursday, November 1
1. PSfrag
Additive-noise
channel with path gain. Consider the additive noise channel shown in the figure
replacements
below, where X and Z are zero mean
EE 278
Statistical Signal Processing
October 11, 2007
Handout #4
Homework #3
Due Thursday, October 18
1. Family planning. Alice and Bob choose a number X at random from the set cfw_2, 3, 4 (so the
outcomes are equally probable). If the outcome is X = x, t
EE 278
Statistical Signal Processing
Thursday, November 8, 2007
Handout #14
Homework #6
Due Thursday, November 15
1. Gaussian random vector Suppose X N (, ) is a Gaussian random vector with
1 1 0
1
= 5 and = 1 4 0 .
0 0 9
2
a. Find the pdf of X1 .
b. F
EE 278
Statistical Signal Processing
Thursday, November 15, 2007
Handout #17
Homework #7
Due Thursday, November 29
1. Vector CLT. The signal received over a wireless communication channel can be represented by
two sums
n
n
1 X
1 X
X1n =
Zj cos j and X2n
EE 278
Statistical Signal Processing
October 18, 2007
Handout #6
Homework #4
Due Thursday, October 25
1. Two envelopes. A fixed amount a is placed in one envelope and an amount 5a is placed in the
other. One of the envelopes is opened (each envelope is eq
EE 278
Statistical Signal Processing
September 27, 2007
Handout #1
Homework #1
Due Thursday, October 4
You can either hand the assignment to me after class or drop it in the
Homework In box in the EE 278 drawer of the class file cabinet on the
2nd floor o