Homework 2 Solutions 1. It is not reasonable to think that the probability of rain sometime on the weekend is 50%. This number is obtained by treating the events (rain on Saturday) and (rain on Sunday) as mutually exclusive, but there is no evidence to su
Helms chapter 2 teaching notes
Dean P Foster
September 21, 2011
Administrivia
Review of state so far
Repeat rules, remind of continuity.
Bonferroni / Bools inequality
P (Ai )
P (Ai )
1/2 of continuity
What we gain: most distributions are singular
So w
Helms 1 teaching notes
Dean P Foster
October 19, 2011
Administrivia
Grading: midterm and nal
Homework is for discussion and learning. (Ill collect it so you
can have it correctedbut it is for your benit.)
So well discuss them in class also.
Two books:
Final Exam
December 10, 2009
1. (15 pts) Suppose X, Y are random variables with the following joint distribution:
f (x, y ) =
(a) Are X and Y independent? cfw_answer:
xy
0
0 < x < 1, 0 < y < 2
otherwise
Yes. Follows from: f (x, y ) = fX (x)fY (y ).
(b) Wh
Probability Final
You may use your sheet from the midterm and a new sheet of notes. No calculators, cell phones, PDAs,
laptops, or HAL 9000s. Show your reasoning. Dont just give the answer.
1. A box contains two gold balls and two clay balls. You are allo
Chapter Game Theory
Dean P Foster
November 21, 2011
1
Administrivia
No class wednesday.
2
Story: Philosphers
3
Common knowledge
A is common knowledge if A F and A G .
Set of such sets is a sigma-algebra
It is all facts everyone knows
Sequential aproa
Chapter from Helms teaching notes
Dean P Foster
September 28, 2010
Administrivia
No hard copy, it is on the web.
It should be available from Dover Publications soon (I couldnt nd
it in 2009).
Homework questions?
Cute problem
Spike and slab in statistics r
Chapter 10.3: CLT
Dean P Foster
September 27, 2010
1
Administrivia
By special request: Homework due friday.
2
Story: Integration
Reimann sum
Lebegue sum
Henstock-Kurzweil integral:
I say, the integral is v
You say, nothat is o by
I say, no, any Rei
Chapter 10: Generating functions
Dean P Foster
September 27, 2010
1
Administrivia
Collect homework. next one due friday.
2
Formal exponentials
What is ex ? Many denitions.
One nice one is as a power series: ex = 1+x+x2 /2+ +xi /i!+ .
Only need three t
Chapter 9: CLT
Dean P Foster
November 7, 2011
1
Administrivia
Why does the normal appear?
Key fact: Normal plus normal is normal.
true for other things (Binomial, Poisson, Gamma, even Cauchy)
but this one is MORE true
The only one which has (X +Y )/ 2
Chapter 8: Law of Large numbers
Dean P Foster
September 27, 2010
1
Administrivia
Yes, we have class next wed
Next homework can be delayed if you like
But there will be one the next weekend. So dont delay very long!
2
All shapes look the same
Tomography
Chapter 8 intro
Dean P Foster
September 27, 2010
Administrivia
Markovs inequality: why
Denition of expectation comes from probability.
How do you compute a probability from an expectation?
working backwardslike division
Expectation is linear, probabilit
Chapter 8: Law of Large numbers
Dean P Foster
September 27, 2010
Administrivia
Homework questions?
Patience
Keynes: In the long run, were all dead.
Singularity: Maybe.
If M = 6, the universe will crush itself in 15 billion years.
If M > 1, sometime we wi
Chapter 7.1 teaching notes
Dean P Foster
September 27, 2010
Administrivia
Homework questions?
Many faces of conditional expectation
There are many ways to dene conditional expectation:
E (Y |X = x) =
x xf (Y
x xP (Y
= x|X = x) (or E (Y |X = x) =
= x|X =
Helms chapter 2 teaching notes
Dean P Foster
September 12, 2011
Administrivia
Homework questions?
Problem statement
Three views:
Bananch Algebra (too much math)
* Language: X, Y, Z, RVs. E (X ), X + Y, Y Z
Discrete sample space i.e. balls in Urns (cant
Helms chapter 2 teaching notes
Dean P Foster
September 21, 2011
Administrivia
Homework questions?
Due Friday
Review: What can measure theory do?
So whats it all good for?
We can do spike and slab now
Same theory for both discrete and continuous.
But
Homework 8 Solutions
1. Ross 6.27 p. 316
Let X ~ U (0, 1) and Y ~ Exp(1) be independent random variables. Then the joint density of X
and Y is f ( x, y ) f X ( x) f Y ( y ) (1)(e y ) e y
a) If Z X Y , the cdf of Z is
FZ (a) P ( X Y a )
a
a a x y
e dydx 1
Homework 6 Solutions
1. Ross 5.5 p. 247
The probability of the gasoline supply being exhausted for a tank capacity c is obtained by
integrating f (x ) from c to 1. Here, we want the probability of the supply exhausted to be 0.01;
hence, we must choose c s
Homework 1 Solutions 1. Ross 1.1 p. 16 a) Since repeats are allowed in this scenario, there are 26 possibilities are each of the first 2 places and 10 possibilities for each of the remaining 5 places; hence, there are 26 2 10 5 = 67,600,000 different 7-pl
Chapter 4.2 teaching notes
Dean P Foster
October 13, 2010
Administrivia
Exam 1 weeks from today
Continuous conditional probability
Table form of random variables:
Do Age (in decades) vs weight (in stones)
Question: What is average age for 40 year old?
Chapter 4.1 teaching notes
Dean P Foster
October 6, 2010
Administrivia
Homework: Asymptotics #19.
Conditional probability as counts
Life table
P(X > 80) = 57062
P(X > 60) = 89835
P(X > 80 | X > 60) = .63 (note > P(X > 80)
Story time: life tables
How
Chapter 3.3 teaching notes
Dean P Foster
November 9, 2011
Administrivia
homework due next tuesday and following tuesday
Read section 3.1 and 3.2 on your own if you have done so already
We will be reading through 3.3 carefully!
Read Pitman chapter 2 (I
Chapter 3.2 teaching notes
Dean P Foster
September 28, 2010
Administrivia
Homework questions?
Game: what is the most important theorem in all of mathematics?
Game we played at the mathematics department:
Intromediate value theorem
central limit theorem
Chapter 3.1 teaching notes
Dean P Foster
September 27, 2010
Administrivia
Homework questions? (keep it short)
Intro story
Think back to 1982, what were you doing? Yikes! Nothing?
I was playing bridge, yes, like your grandparent.
I wanted distribution o
Chapter 2.1 teaching notes
Dean P Foster
September 27, 2010
Administrivia
Homework due friday in box outside my oce.
needles on the pavement
continuous math is easier
calculus
diferential equations
approximations
What about a continuous probability?
Chapter 1.2 teaching notes
Dean P Foster
September 27, 2010
Administrivia
homework questions?
Random variables
We have, the idea of X , and Y that are random variables. We can do
X +Y
X Y
3X
X2
X3
eX
at least conceptually. But we dont know what a Random
Chapter 1.1 teaching notes
Dean P Foster
September 27, 2010
Administrivia
See web page. (google, Dean Foster, follow links.)
Summary:
midterm + nal
Lots of homework
Free book (on line and editable, like the wiki)
Hence suggestions for book are in orde
Homework 6 Solutions
6.2
6.3
14. Break up the integral into three parts: from (-infinity, -1), (-1,+1) and (+1,+infinity). Use the hint on
the first and third integrals. Then recombine to get the integral with x^2 as the integrand.
7.1
7.2
8.1
8.2
Helms Section 4.2
Dean P Foster
October 12, 2011
Administrivia
Homework questions? (Due Friday)
Alternative theory
Let X, Y, Z be members of a set X
If X, Y X then X + Y and XY are in X
For all X X we have E (X ) dened
There exists a random variable
Chapter 6.3 teaching notes
Dean P Foster
September 27, 2010
Administrivia
collect homework
Story 1: distances for cities
How similar are Philly and SF?
List lots, then no two cities are close
List too few (say just latitude) and the NYC and madrid are