Notes 12: Sampling Distributions and the Central Limit Theorem
Associated Reading: Wackerly 7, Chapter 7, Sections 1-4
This chapter will conclude the discussion of functions of random variables that began in Chapter 5, and lay the
last groundwork that y
VERVRDH 'M
1. (6 points) You repeatedly toss two fair twelve-sided dice. What is the expected number of throws you need
to make until the rst time you observe the same number on each die face (i.e., until you see a 2 and a
2 or a 5 and a 5, etc.)? For ful
VEQSiON M
1. (8 points) You are given two r.v.s, Y1 and Y2, whose variances are V[Y1] = V[Y2] = 3 and whose covariance
is 2. You dene a new r.v. U = 3Yl Y2. What is V[U]?
VEUE=QTiQ = (5 4) 3 I (3 = (a 4) 92:;
7. 3) "I (9-3: 3)
= 'Zl-B =ll8l
om 013va + qfv
Notes 8: The Multivariate Normal
Associated Reading: Wackerly 7, Chapter 5, Section 10
In this notes set, we will introduce the multivariate normal distribution. Wackerly 7 barely touches upon it, by
only writing down the pdf for the bivariate normal; the
WERS'KDN B
1. (6 points) You repeatedly toss two fair eightsided dice. What is the expected number of throws you need
to make until the rst time you observe the same number on each die face (i.e., until you see a 2 and a
2 or a 5 and a 5, etc)? For full c
Notes 7: Multivariate Distributions: Expected Value
Associated Reading: Wackerly 7, Chapter 5, Sections 5-8 and 11
In Chapters 3 and 4, you were introduced to the expected value operator, which takes the weighted average of a
random variable (or a functio
MIL
Notes 9: Distributions of Functions of Random Variables
Associated Reading: Wackerly 7, Chapter 6, Sections 1-4
Lets start by establishing the point of this chapter. Youve conducted an experiment which yields observed data
Y1, - - - ,Y,
M1
Notes 13: Some Basic Concepts from Information Theory
Associated Reading: e.g., Pattern Recognition and Machine Learning by Bishop, pp. 48-58
Information theory, which grew out of Claude Shannons 1948 journal article A Mathematical Theory o
Mfl.
Notes 5: Commonly Used Continuous Distributions
Associated Reading: Wackerly 7, Chapter 4, Sections 4-8
In these notes, we shift from talking about discrete distributions to continuous distributions, highlighting the
uniform, normal, gamm
Notes 6: Multivariate Distributions
Associated Reading: Wackerly 7, Chapter 5, Sections 1-4
Up to now, weve concentrated on mrivan'ate probability distributions, i.e., distributions dened along any one
axis (usually denoted y) in Cartesian space. So now w
M1
Notes 11: Order Statistics
Associated Reading: Wackerly 7, Chapter 6, Section 7
The background: you sample n .119 r.v.s, denoted cfw_Y1, Y2, . . . , Y , from some distribution. There are various
statistics (which, you remember, are simply
Notes 15: A Very Short Introduction to Markov Processes
Associated Reading: e.g., Introduction to Probability by Bertsekas & Tsitsiklis, pp. 313-321
Lets make a series of observations of the weather: _ -
Afs Lt etc l'.'\ "W
X03X15X25' an " seeypcwce
The r
Notes 14: Some Basic Concepts from Probabilistic Graphical Modeling
Associated Reading: e.g., Pattern Recognition and Machine Learning by Bishop, pp. 359-383
(Note, however, that we will not be following the text in depth, so skimming the reading is ne.)
\j ERSSN Q
1. (8 points) You are given two r.v.s, Y1 and Y2, whose variances are V[Y1] = V[Y2] = 3 and whose covariance
is 2. You dene a new r.v. U = 2Y1 Y2. What is V[U]?
V10] = QTEQ = (7- ") 3 Z>(2): (2 -l) (0-17.4
2. 3 I 9-521
': 3-\ = i? I
On: \IYU] '
VERSloN B
1. (8 points overall) You are given the following directed acyclic graph:
(a) (4 points) Write the joint pmf p(a, b, c, d, e) as a product of four conditional and one unconditional pmfs.
(b) (4 points) Is D JL CIA, B, E? Show all reasoning for f
\I'EQ 310K: 3
1. (8 points) You are given two r.v.s, Y1 and Yg, whose variances are V[Y1] = V[Y2] = 4 and whose covariance
is 2. You dene a new r.v. U = Y1 2Y2 What is V[U]?
$03 = GTE = ( 'Z5Ki 1)(\1)=(\*z) -4 :o
3'. LI - 2_<5='-(-:
Om E]
on: viu] Via] +
VER5i O N P:
1. (6 points) You repeatedly toss two fair sixsided dice. What is the expected number of throws you need to
make until the rst time you observe the same number on each die face (i.e., until you see a 2 and a 2
or a 5 and a 5, etc.)? For full
VERSiON Fl
1. (8 points overall) You are given the following directed acyclic graph:
(a) (4 points) Write the joint pmf p(a, b, c, d, e) as a product of four conditional and one unconditional pmfs.
(b) (4 points) Is D L E IA, B, C? Show all reasoning for
VERSlOM M
l. (8 points overall) You are given the following directed acyclic graph:
0
a,
(a) (4 points) Write the joint pmf p(a, b, c, d, e) as a product of four conditional and one unconditional pmfs.
(b) (4 points) Is D JL E IA, B, C? Show all reasoning