UCSD ECE153
Prof. Young-Han Kim
Handout #7
Wednesday, January 21, 2015
Weekly Exercise Set #3
1. Let A be a nonzero probability event A. Show that
(a) P(A) = P(A|X x)FX (x) + P(A|X > x)(1 FX (x).
P(A|X x)
FX (x).
(b) FX (x|A) =
P(A)
2. Joint cdf or not. C
UCSD ECE250
Prof. Young-Han Kim
Handout #18
Wednesday, February 17, 2016
Solutions to Midterm Examination
(Prepared by TA Shouvik Ganguly)
There are 4 problems, each problem with multiple parts, each part worth 10 points. Your
answer should be as clear an
UCSD ECE250
Prof. Young-Han Kim
Handout #18
Wednesday, March 4, 2015
Exercise Set #7
1. Symmetric random walk. Let Xn be a random walk dened by
X0 = 0,
n
Xn =
Zi ,
i=1
1
where Z1 , Z2 , . . . are i.i.d. with Pcfw_Z1 = 1 = Pcfw_Z1 = 1 = 2 .
(a) Find Pcfw_X
UCSD ECE250
Prof. Young-Han Kim
Handout #17
Wednesday, March 4, 2015
Solutions to Exercise Set #6
(Prepared by TA Fatemeh Arbabjolfaei)
1. Linear estimator. Consider a channel with the observation Y = XZ, where the signal X and
2
the noise Z are uncorrela
UCSD ECE250
Prof. Young-Han Kim
Handout #14
Wednesday, February 11, 2015
Midterm Examination
(Total: 100 points)
There are 4 problems, each problem with multiple parts, each part worth 10 points. Your
answer should be as clear and readable as possible. Pl
UCSD ECE250
Prof. Young-Han Kim
Handout #16
Wednesday, February 18, 2015
Exercise Set #6
1. Linear estimator. Consider a channel with the observation Y = XZ, where the signal X and
2
the noise Z are uncorrelated Gaussian random variables. Let E[X] = 1, E[
UCSD ECE250
Prof. Young-Han Kim
Handout #19
Wednesday, March 11, 2015
Solutions to Exercise Set #7
(Prepared by TA Fatemeh Arbabjolfaei)
1. Symmetric random walk. Let Xn be a random walk dened by
X0 = 0,
n
Zi ,
Xn =
i=1
1
where Z1 , Z2 , . . . are i.i.d.
UCSD ECE250
Prof. Young-Han Kim
Handout #12
Monday, February 9, 2015
Solutions to Exercise Set #5
(Prepared by TA Fatemeh Arbabjolfaei)
1. Neural net. Let Y = X + Z, where the signal X U[1, 1] and noise Z N (0, 1) are
independent.
(a) Find the function g(
UCSD ECE250
Prof. Young-Han Kim
Handout #13
Monday, February 9, 2015
Midterm Review
(Prepared by TA Fatemeh Arbabjolfaei)
1. Order statistics.
Let X1 , X2 , X3 be independent and uniformly drawn from the interval [0, 1]. Let Y1 be the
smallest of X1 , X2
ECE 250 Midterm
November 7, 2014
SOLUTION
1. Let Xk, k = 1, 2, . , be independent and identically-distributed with common density
fx(x)=cfw_e-x, x>O
O,x<O
ConsiderthesumZn
ixk.
=
k=I
ElX)o.J = S~ ~ t
0
ProvethatforanyE>O, lim P(l Zn -11
n~oo
n
~&)=0.
An a
UCSD ECE250
Prof. Young-Han Kim
Handout #10
Wednesday, February 4, 2015
Solutions to Exercise Set #4
(Prepared by TA Fatemeh Arbabjolfaei)
1. Two envelopes. An amount A is placed in one envelope and the amount 2A is placed in
another envelope. The amount
UCSD ECE250
Prof. Young-Han Kim
Handout #11
Wednesday, February 4, 2015
Exercise Set #5
1. Neural net. Let Y = X + Z, where the signal X U[1, 1] and noise Z N(0, 1) are
independent.
(a) Find the function g(y) that minimizes
MSE = E (sgn(X) g(Y )2 ,
where
UCSD ECE250
Prof. Young-Han Kim
Handout #22
Wednesday, February 18, 2015
Solutions to Final Examination
(Prepared by TA Fatemeh Arbabjolfaei)
1. Drawing balls without replacement (20 pts). Suppose that we have an urn containing one
red ball and n 1 white
UCSD ECE250
Prof. Young-Han Kim
Handout #16
Saturday, February 13, 2016
Solutions to Exercise Set #6
(Prepared by TA Shouvik Ganguly)
1. Packet switching. Let N be the number of packets per unit time arriving at a network
switch. Each packet is routed to
UCSD ECE250
Prof. Young-Han Kim
Handout #3
Wednesday, January 7, 2015
Weekly Exercise Set #1
1. Using the axioms of probability, show that A B implies that P(A) P(B).
2. Independence. Show that the events A and B are independent if P(A|B) = P(A|B c ).
3.
UCSD ECE250
Prof. Young-Han Kim
Handout #9
Wednesday, January 28, 2014
Exercise Set #4
1. Two envelopes. An amount A is placed in one envelope and the amount 2A is placed in
another envelope. The amount A is xed but unknown to you. The envelopes are shued
UCSD ECE250
Prof. Young-Han Kim
Handout #5
Wednesday, January 14, 2015
Weekly Exercise Set #2
1. Jurors fallacy. Suppose that P(A|B) P(A) and P(A|C) P(A). Is it always true that
P(A|B, C) P(A) ? Prove or provide a counterexample.
2. Polyas urn. Suppose we
UCSD ECE250
Prof. Young-Han Kim
Handout #6
Wednesday, January 21, 2015
Solutions to Exercise Set #2
(Prepared by TA Fatemeh Arbabjolfaei)
1. Jurors fallacy. Suppose that P (A|B) P (A) and P (A|C) P (A). Is it always true that
P (A|B, C) P (A) ? Prove or p
UCSD ECE250
Prof. Young-Han Kim
Handout #4
Wednesday, January 14, 2015
Solutions to Exercise Set #1
(Prepared by TA Fatemeh Arbabjolfaei)
1. Using the axioms of probability, show that A B implies that P(A) P(B).
Solution: Since (B Ac ) and (B A) = A are d
UCSD ECE 250
Prof. Young-Han Kim
Handout #8
Wednesday, January 28, 2015
Solutions to Exercise Set #3
(Prepared by TA Fatemeh Arbabjolfaei)
1. Let A be a nonzero probability event. Show that
(a) P(A) = P(A|X x)FX (x) + P(A|X > x)(1 FX (x).
(b) FX (x|A) =
P
UCSD ECE250
Prof. Young-Han Kim
Handout #20
Wednesday, March 11, 2015
Final Review
(Prepared by TA Fatemeh Arbabjolfaei)
1. Random-delay mixture.
Let cfw_X(t), < t < , be a zero-mean wide-sense stationary process with autocorrelation
function RX ( ) = e|
UCSD ECE250
Prof. Young-Han Kim
Handout #21
Wednesday, February 18, 2015
Final Examination
There are 6 problems, each problem with multiple parts, each part worth 10 points. Your answer
should be as clear and readable as possible. Justify any claim that y
ECE 250 FALL QUARTER 2015
ELECTRICAL & COMPUTER ENGINEERING
Course:
ECE 250 Random Processes
URL:
https:/sites.google.com/a/eng.ucsd.edu/ece-250-f2015/
Text:
H. Stark & J. W. Woods, Probability and Random Processes with
Applications to Signal Processing,
UCSD ECE250
Prof. Young-Han Kim
Handout #6
Wednesday, January 21, 2015
Solutions to Exercise Set #2
(Prepared by TA Shouvik Ganguly)
1. Jurors fallacy. Suppose that P (A|B) P (A) and P (A|C) P (A). Is it always true that
P (A|B, C) P (A) ? Prove or provid
UCSD ECE 250
Prof. Young-Han Kim
Handout #8
Wednesday, January 28, 2015
Solutions to Exercise Set #3
(Prepared by TA Shouvik Ganguly)
1. Nonlinear processing. Let X Unif [1, 1]. Define the random variable
2
X + 1, if |X| 0.5
Y =
0,
otherwise.
Find and sk
UCSD ECE250
Prof. Young-Han Kim
Handout #10
Thursday, February 4, 2016
Solutions to Exercise Set #4
(Prepared by TA Shouvik Ganguly)
1. Optical communication channel. Let the signal input to an optical channel be given by
1 with probability 21
X=
10 with
UCSD ECE250
Prof. Young-Han Kim
Handout #13
Wednesday, February 8, 2016
Solutions to Practice Midterm
There are 4 problems, each problem with multiple parts, each part worth 10 points. Your
answer should be as clear and readable as possible. Please justif
UCSD ECE250
Prof. Young-Han Kim
Handout #26
Tuesday, March 22, 2016
Solutions to Final Examination
(Prepared by TA Shouvik Ganguly)
There are 4 problems, each problem with multiple parts, each part worth 10 points. Your
answer should be as clear and reada
UCSD ECE250
Prof. Young-Han Kim
Handout #19
Wednesday, February 24, 2016
Exercise Set #7
1. Minimum waiting time. Let X1 , X2 , . . . be i.i.d. exponentially distributed random variables
with parameter , i.e., fXi (x) = ex , for x 0.
(a) Does Yn = mincfw_
UCSD ECE250
Prof. Young-Han Kim
Handout #2
Wednesday, January 6, 2016
Solutions to ECE250 Aptitude Test
1.
X
rn =
n=0
1
.
1r
Solution: Let
S = 1 + r + r2 + .
(The series converges since 0 < r < 1.) Then
rS = r + r 2 + r 3 + .
Taking the difference between
UCSD ECE250
Prof. Young-Han Kim
Handout #22
Wednesday, March 2, 2016
Practice Final Examination (Winter 2015)
There are 6 problems, each problem with multiple parts, each part worth 10 points. Your answer
should be as clear and readable as possible. Justi
UCSD ECE250
Prof. Young-Han Kim
Handout #25
Wednesday, March 16, 2016
Final Examination
(Total: 170 points)
There are 4 problems, each problem with multiple parts, each part worth 10 points. Your answer
should be as clear and readable as possible. Justify