Stat 421 Solutions for Homework Set 5
Page 129 Exercise 5: Suppose that the joint p.d.f. of two random variables X and Y is as follows:
cfw_
c(x2 + y) for 0 y 1 x2 , 1 < x < 1
f (x, y) =
0
otherwise.
Determine (a) the value of the constant c; (b) Pr(0 X 1
Notes for Chapter 2 of DeGroot and Schervish
Conditional Probabilities
Sometimes if we are given additional information we can reduce our sample space.
If we know an event F occurs, then the probability of another event E occurring may
change.
Ex. It snow
PROBLEM SET 1
1. Exercise 90 (p71).
2. Exercise 93 (p71).
3. Exercise 94 (p71).
4. Exercise 100 (p77).
5. Exercise 101 (p77).
6. Exercise 65 (p109).
7. Exercise 69 (p109).
8. P (A) =
1
3
and P (B c ) = 14 , can A and B be disjoint? Explain.
9. Suppose tha
Counting rules useful in probability, 2.4 and 2.5
Grethe Hystad
February 6, 2011
Grethe Hystad
Counting rules useful in probability, 2.4 and 2.5
Counting rules useful in probability, 2.4
Theorem
Fundamental Principle of Counting:
If the first task of an e
PROBLEM SET 2-1
For this problem set assume all the integrals exist.
1. Prove the following; for any two random variables X, Y ,Z
(a) E[X] = E[E[X | Y ].
(b) E[X | Y ] = E[X] if X and Y are independent.
(c) E[Xg(Y ) | Y ] = g(Y )E[X | Y ], in particular,
PROBLEM SET 1
1. Exercise 90 (p71).
2. Exercise 93 (p71).
3. Exercise 94 (p71).
4. Exercise 100 (p77).
5. Exercise 101 (p77).
6. Exercise 65 (p109).
7. Exercise 69 (p109).
8. P (A) =
1
3
and P (B c ) = 14 , can A and B be disjoint? Explain.
9. Suppose tha
Broadband Wireless Communication Systems
Lecture 2. Statistical Multipath
Channel Model
Digital Communications Lab.
Fading channel
Digital Communications Lab.
Fading channel
Large-scale fading
Average signal power attenuation due to motion over large ar
Special Topics in Communication Systems
Week 2. Review on Random
Process (2)
Digital Communications Lab.
Specification of Random Processes
The probability distribution for random vectors
PX ( F ) P( X1 ( F ) Pcfw_ | X( ) F
P(cfw_ | ( X 0 ( ),
, X k 1 (
Broadband Wireless Communication Systems
Week 6-2. Adaptive modulation
and coding
Digital Communications Lab.
Adaptive Modulation
Variable-rate variable-power MQAM system
Shannon capacity places no restriction on the complexity or delay of
the multiplexe
Special Topics in Communication Systems
Week 1. Review on Random
Process (1)
Digital Communications Lab.
Set Theory and Set Operations
Set
Group or collection of the elements
Point set : a set with only one element
Empty set : a set with no element
S
Broadband Wireless Communication Systems
Week 4-2. Digital transmission over
wireless channels
Digital Communications Lab.
Signal space
Consider a set of signals (waveforms) cfw_xi(t) defined on [0,T]
Inner product
T
Norm
x1 (t ), x2 (t ) x1 (t ) x2* (t
Special Topics in Communication Systems
Week 3. Review on Random
Process (3)
Digital Communications Lab.
Linear System
In general, a system L is a mapping of an input time function x=cfw_x(t);
tI into output function L(x)=y=cfw_y(t); tI
The system is ca
Broadband Wireless Communication Systems
Lecture 3. Background on
Information Theory
Digital Communications Lab.
Background of information theory
Entropy
A measure of the uncertainty of a random variable.
Def. Entropy :
The entropy of a discrete rando
Broadband Wireless Communication Systems
Week 7. Capacity with CSIR and
one-bit feedback
Digital Communications Lab.
Channel and System Model
Scenarios for the knowledge of the channel side
information (CSI)
Channel distribution information (CDI)
The d
Broadband Wireless Communication Systems
Week 6-1. Wireless Channel
Capacity
Digital Communications Lab.
Capacity of AWGN Channel
Differential entropy
For a continuous random variable X,
h X
f x log f x dx
Theorem
For a given variance, differential en
Broadband Wireless Communication Systems
Lecture 9. Frequency Selective
Channel and OFDM Principle
Digital Communications Lab.
Signal design for band-limited channels
Low-pass equivalent model
t kT 0
GR ( f )
H( f )
y (t )
r (t )
s(t ) I n gT (t nT )
n(t
Broadband Wireless Communication Systems
Lecture 8. MIMO channel capacity and
ST receivers
Digital Communications Lab.
MIMO channel capacity
SISO channel
h ~ CN (0,1) : A zero-mean complex Gaussian noise with variance 1.0.
SISO channel
y(t ) hx(t ) n(t
Broadband Wireless Communication Systems
Lecture 1. Wireless Channels,
Pathloss, and Shadowing
Digital Communications Lab.
Channel Characteristic
Wireless channel
Using Antenna (>0.1l)
1MHz 0.1l=30m
Transmission modes
Ground wave
As waveguided for VL
Steel Framed structures
Steel frame is a building technique with skeleton frame of vertical steel columns and horizontal
I-beams, constructed in a rectangular grid to support the floors, roof and walls of a building
which are all attached to the frame.
Fi
Chapter 11. Fourier Analysis
1
11.1 Fourier Series
Denition (Periodic Functions)
A function f (x) is called a periodic function if p > 0
(called a period of f ) such that f (x + p) = f (x) for all x.
Ex. periodic : sin 2x, cos 3x, sin 2x + 3 cos 3x,
not
Chapter 20. Numerical Linear Algebra
1
20.1 Linear Systems: Gauss Elimination
A linear system of n equations in n unknowns
E1 : a11x1 + a12x2 + + a1nxn = b1
E2 : a21x1 + a22x2 + + a2nxn = b2
En : an1x1 + an2x2 + + annxn = bn
A single vector equation
Ax
Chapter 12. Partial Dierential Equations(PDEs)
1
12.1 Basic Concept of PDEs
Partial dierential equations(PDEs)
- an equation involving one or more partial derivatives of an unknown function that depends on two or more variables.
Ex. When u = u(x, y), ux
Chapter 23. Graphs. Combinatorial Optimization
1
23.1 Graphs and Digraphs
Denition (Graph)
A graph G consists of two nite sets (sets having nitely many elements),
a set V of points, called vertices, and a set E of connecting lines, called
edges, such that
Chapter 10. Vector Integral Calculus. Integral Theorems
1
10.1 Line integrals
Denition (Line integral)
Let C be a (piecewise smooth) curve
r(t) = [x(t), y(t), z(t)] = x(t)i + y(t)j + z(t)k (a t b),
and let F be a vector function over C defined by
F(r) = F
Chapter 9. Vector Dierential Calculus, Grad, Div, Curl
1
9.7 Gradient of a Scalar Field
Definition (Gradient)
Let f (x, y, z) be dierentiable. Then the gradient of
f (x, y, z) is dened
[ by the vector
] function
f
f
f
f f f
,
,
=
i+
j+
k
gradf = f =
x y z
#1.
,
#2. (a)
cos , where
sin
(b) cos sin
(c)
#3.
#4.
#5. Hint :By Stokes theorem,
.
So we only need to show that .
#6. (a) 11.9 #3(a)
(b)
#7. (a) cos cos cos cos
(b)
#8. (a)
sin cos sin
1
:
(4)
:
:
1. (5pts) Find a unit vector in the direction in
which decreases most rapidly
(b) (2pts) For arbitrary point in , find the
normal vector at .
at , and find the rate of change of
at in that direction.
(c) (4pts) Find the area of the surfac
Homework
23.1
1. Find the adjacency matrix of the given graph or digraph.
(a)
(b)
(c)
1
(d)
2. Incidence matrix
cfw_ B of a graph is B = [bjk ], where
1 if vertex j is an endpoint of edge ek
bjk =
0 otherwise
of a digraph is B
= [bjk ], where
Incidence