EE 131A
Homework #8
Due March 4th
Winter 2015
K. Yao
Watch Lec15-1, 15-2, 16-1, and 16-2 and read relevant pages in the textbook (Leon-Garcia 3rd edition).
Lectures
Lec15-1
Lec15-2
Lec16-1
Lec16-2
Textbook Pages
Pp. 67-70;194-196
None
Pp. 462-468
Pp. 334-
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 10-1
UCLA EE131A (KY)
1
Expectation of a function g(X) of X (1)
So far, we have considered the expectation Ecfw_. of
g(X) = X to yields
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 11-1
UCLA EE131A (KY)
1
Jointly pdf and marginal pdf (1)
Let FXY(x,y) be the joint cdf of the joint rvs (X,Y),
where X and Y can be con
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 7-2
UCLA EE131A (KY)
1
Known continuous rvs: 1. Uniform rv
1. Uniform rv - A uniform rv X characterizes a
random experiment with equal l
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 10-2
UCLA EE131A (KY)
1
Jointly distribution of two rvs (1)
For a single rv X, it is fully characterized by its cdf
FX(x) or the pdf fX
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 8-2
UCLA EE131A (KY)
1
Function of a rv (1)
A rv is a function defined on the sample space S.
Consider a real-valued function g(x) with
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 9-2
UCLA EE131A (KY)
1
Second Moment (1)
The expectation of a rv X yields the mean .
The second moment m2 is the expectation of X2
def
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 9-1
UCLA EE131A (KY)
1
Expectation (averaging) (1)
The cdf/pdf of a rv X completely characterizes the rv.
Often we may want to have a s
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 8-1
UCLA EE131A (KY)
1
non negative
Gaussian (Normal) rv (1)
Gaussian rv - A Gaussian rv X with two parameters
( , 2) has a pdf given by
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 3 - 2
UCLA EE131A (KY)
1
Lecture 3 -2: Axioms of probability (1)
The theory of modern probability was
developed formally by A.N. Kolmog
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 7-1
UCLA EE131A (KY)
1
Known discrete rvs: 1. Bernoulli rv
1. Bernoulli rv - A Bernoulli rv X characterize a
random experiment with two
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 6-1
UCLA EE131A (KY)
1
Random Variable (1)
So far, we have dealt with elementary outcomes and
collection of elementary outcomes called
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 6-3
UCLA EE131A (KY)
1
Continuous random variable (1)
discrete
A continuous random variable X is defined as a
rv whose cdf F(x) is cont
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 6-2
UCLA EE131A (KY)
1
CDFX <= x
Cumulative Distribution Function (5)
Ex. 1. Toss a coin. Sample space S = cfw_H, T; P(H) = 0.6
and P(T)
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 5-2
UCLA EE131A (KY)
1
Binomial Probability Law (1)
Consider a binary-valued experiment with outcomes
called a success (S) or a failure
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 5-1
UCLA EE131A (KY)
1
Independence of Events (1)
Consider two events A and B. If the occurrence of
event B does not affect the probabil
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 11-2
UCLA EE131A (KY)
1
Independence of two rvs (1)
Review: Two events A and B are independent if
P(AB) = P(A)P(B) .
Now, we want to c
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 13-1
UCLA EE131A (KY)
1
Distribution and pdf of (X+Y) (1)
When we have two rvs, X and Y, often they may
interact in the form of (X + Y)
EE 131A Probability
Professor Kung Yao
Electrical Engineering Department
University of California, Los Angeles
Lecture 12-2
UCLA EE131A (KY)
1
Covariance (1)
The covariance of rvs X and Y provides a statistical
average relationship between (X-X) and (Y-Y
EE 131A
Homework #6
Due Feb. 23rd
Winter 2015
K. Yao
Read Leon-Garcia (3rd edition), pp. 155-180; 255-278.
1. Let S be the speed of a randomly selected molecule in a gas. According to the kinetic
theory of gases, S has the Maxwell pdf of fS (s) = as2 exp(
EE 131A
Homework #7
Due Feb. 25th
Winter 2015
K. Yao
Read Leon-Garcia (3rd edition), pp. 115-116; 163-165; 181-189; 233-284.
1. Let X and Y have a joint pdf of f (x, y) = e(x+y) , x 0, y 0. Find P (X Y 2).
Hint: Sketch out the region in the x-y plane that
EE 131A
Homework #4
Due Feb. 4th
Winter 2015
K. Yao
Read Leon-Garcia (3rd edition), pp. 146-152; 163-180
1. The Rayleigh random variable has c.d.f.
FR (r) =
(
0,
1 e
r 2 /2
2
,
r<0
r 0,
Find P [ R 2 ] and P [R > 3 ]. Find the p.d.f. of this r.v.
2. A r.v.
EE 131A
Homework #5
Due Feb. 11th
Winter 2015
K. Yao
Read Leon-Garcia (3rd edition), pp. 148-188; 233-254.
1. Problem 4.63 (p. 221).
2. A limiter Y = g(X) is shown in Fig. P4.2 (p. 219).
a. Find the cdf and pdf of Y in terms of the cdf FX (x) and pdf fX (
EE 131A
Homework #3
Due Jan. 28th th
Winter 2015
K. Yao
Read Leon-Garcia (3rd edition), pp. 96-104; 141-146,
1. A batch of 500 containers of frozen orange juice contains 5 that are defective. Two are
selected randomly one after the other without replaceme
EE 131A
Homework #3
Solution
Winter 2015
K. Yao
1. a. 4/499 = 0.008
b. (5/500)(4/499)=0.00008
c. (495/500)(494/499)=0.98
2. a. To check for independence between A and B, verify whether P (B|A) = P (B)? P (B|A) =
4/499 = 0.008; P (B) = (495/500)(5/499) + (
EE 131A
Midterm Exam
Winter 2014
(60 minutes/25 pts - An 8.5x11 sheet(two-sided) is allowed)
K. Yao
1. An airplane engine fails with a probability of q > 0 . Assume all failures occur independently. Find the probability that:
a. P2 = At least two of the e
EE 131A
Homework #2
Due Jan. 21st
Winter 2015
K. Yao
Read Leon-Garcia (3rd edition), Chap. 2 (pp. 21-58)
1. The professor of a class gave a set of 12 review problems and told the students that the
midterm exam will consists of 6 of these review problems t