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
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 continu
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 b
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
otherwi
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 c
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-al
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
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 int
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 powe
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 th
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
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 ba
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 cru
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)
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 +
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
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 ,
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
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;
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 s
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 r
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 departme
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 bridg
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
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
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,
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 inte
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
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