Math 115 - Team Homework Assignment #9, Winter 2014
Due Date: April 7 or 8, 2014 (Your instructor will tell you the exact date and time.)
Note: All problem, section, and page references are to the course textbook, which is the
6th edition of Calculus: S
Math 115 Second Midterm
March 27, 2014
Name:
EXAM SOLUTIONS
Instructor:
Section:
1. Do not open this exam until you are told to do so.
2. This exam has 11 pages including this cover. There are 11 problems. Note that the problems
are not of equal diculty,
On my honor, as a student, I have neither given
nor received unauthorized aid on this academic work. Signed:
Math 115 Second Midterm
March 27, 2014
Name:
Instructor:
Section:
1. Do not open this exam until you are told to do so.
2. This exam has 11 pages
Math 115 First Midterm
February 11, 2014
Name:
EXAM SOLUTIONS
Instructor:
Section:
1. Do not open this exam until you are told to do so.
2. This exam has 10 pages including this cover. There are 10 problems. Note that the problems
are not of equal diculty
Math 115 - Team Homework Assignment #8, Winter 2014
Due Date: March 20 or 21, 2014 (Your instructor will tell you the exact date and time.)
Note: All problem, section, and page references are to the course textbook, which is the
6th edition of Calculus:
On my honor, as a student, I have neither given
nor received unauthorized aid on this academic work. Signed:
Math 115 First Midterm
February 11, 2014
Name:
Instructor:
Section:
1. Do not open this exam until you are told to do so.
2. This exam has 10 page
Math 115 - Team Homework Assignment #7, Winter 2014
Due Date: March 13 or 14, 2014 (Your instructor will tell you the exact date and time.)
Remember to follow the guidelines from the Doing Team Homework and Team HW Tutorial
links in the sidebar of the c
Math 115 - Team Homework Assignment #6, Winter 2014
Due Date: February 27 or 28, 2014 (Your instructor will tell you the exact date and time.)
Note: All problem, section, and page references are to the course textbook, which is the 6th edition
of Calcul
Math 115 - Team Homework Assignment #3, Winter 2014
Due Date: January 30 or 31, 2014 (Your instructor will tell you the exact date and time.)
Remember to follow the guidelines from the Doing Team Homework and Team HW Tutorial
links in the sidebar of the
Math 115 - Team Homework Assignment #5, Winter 2014
Due Date: February 20 or 21, 2014 (Your instructor will tell you the exact date and time.)
Remember to follow the guidelines from the Doing Team Homework and Team HW Tutorial links in
the sidebar of th
Math 116 Second Midterm
March 24th, 2014
Name:
Instructor:
Section:
1. Do not open this exam until you are told to do so.
2. This exam has 10 pages including this cover. There are 11 problems. Note that the problems
are not of equal diculty, so you may wa
Math 116 Second Midterm
March 24th, 2014
Name:
EXAM SOLUTIONS
Instructor:
Section:
1. Do not open this exam until you are told to do so.
2. This exam has 10 pages including this cover. There are 11 problems. Note that the problems
are not of equal diculty
Math 116 First Midterm
February 10, 2014
Name:
EXAM SOLUTIONS
Instructor:
Section:
1. Do not open this exam until you are told to do so.
2. This exam has 11 pages including this cover. There are 11 problems. Note that the problems
are not of equal diculty
Math 116 First Midterm
February 10, 2014
Name:
Instructor:
Section:
1. Do not open this exam until you are told to do so.
2. This exam has 11 pages including this cover. There are 11 problems. Note that the problems
are not of equal diculty, so you may wa
Conditional Probability
Conditional Probability
Very few experiments amount to just one action with random
Math 425
Introduction to Probability
Lecture 8
outcomes.
Sometimes conditions change before the experiment is
completed.
Some experiments have a mo
Review: Conditioning Rule and Conjunctions
Conditional probability
Math 425
Introduction to Probability
Lecture 9
Denition (Conditioning Rule)
Let E and F be events.
The conditional probability of E given F is dened by:
P(E | F ) =
Kenneth Harris
kaharri@
Overview
Unions and Intersections
! This lecture will focus on computing the probability of arbitrary nite
Math 425
Introduction to Probability
Lecture 7
unions of events
P (E1 E2 . . . En )
even when the events are not mutually inclusive.
The method is c
Review
The probability model
Math 425
Introduction to Probability
Lecture 6
The basis of probability is the experiment:
An experiment is a repeatable procedure that has a a measurable
outcome which cannot be predicted ahead of time.
The probability mode
The probability model
The probability model
Math 425
Introduction to Probability
Lecture 5
The basis of probability is the experiment:
An experiment is a repeatable procedure that has a a measurable
outcome which cannot be predicted ahead of time.
The p
A problem of Probability
A probability problem
Math 425
Introduction to Probability
Lecture 4
Lets begin by gaming and gambling, where the of probability lies.
Problem. Which event is more likely?
1
At least one six on 4 throws of a die.
2
Kenneth Harris
Example: Coin tossing
Example continued
Consider a Bernoulli trials process with IID indicator variables
Math 425
Intro to Probability
Lecture 37
X1 , X2 , . . . denoting whether the trial was a success or failure. Suppose
the probability of success is p
Bounding Probabilities
Chebyshevs Inequality
A measure of the concentration of a random variable X near its
mean is its variance .
Chebyshevs Inequality says that the probability that X lies outside
Math 425
Introduction to Probability
Lecture 36
2
an a
Conditional Expectation continuous
Denition Continuous Case
Let X and Y be jointly continuous random variables with joint
Math 425
Intro to Probability
Lecture 35
density f (x , y ). The conditional probability density function of X given
that Y = y was
Expectation of Products
Expectation of Products
expectations. Consider Xthat an expectation of a product is a product
It is NOT generally true
of
where
Math 425
Introduction to Probability
Lecture 33
P cfw_X = 0 = P cfw_X = 1 = P cfw_X = 1 =
1
3
and dene
Conditional Expectation discrete
Denition Discrete Case
Let X and Y be jointly discreteYrandom variables.cfw_Y =dened the
We
conditional mass of X given that = y (provided P
y > 0) by
Math 425
Introduction to Probability
Lecture 34
pX |Y (x |y )
=
P(X =
Review
Expectation of Sums
The expectation of a sum is a sum of expectations.
Math 425
Introduction to Probability
Lecture 32
Theorem
If X and Y are random variables whose expectations exist and c , d
constants, then
E [cX + dY ] = cE [X ] + dE [Y ].
Ken
Expectation of a Function of Random Variables
Joint Expectation
Let X and Y be continuous (discrete) random variables X and Y
Math 425
Intro to Probability
Lecture 31
with joint density (mass) fX ,Y (x , y ). Let Z be a random variable
determined by X an
Review
Problem: Sitting in a circle
Problem. 5 boys and 5 girls are to be assigned seats so that boys and
girls alternate. How many arrangements are there if they are sat in a
line? How many if they are sat in a circle?
Math 425
Introduction to Probabilit
Introduction One Function of Random Variables
Functions of a Random Variable: Density
Let g(x ) = y be a one-to-one function whose derivative is nonzero
Math 425
Intro to Probability
Lecture 30
on some region A of the real line.
Suppose g maps A onto B ,
Discrete Conditional Probability
Denition: Discrete Conditional Probability
Denition
Let X and Y be discrete random variables on the sample space.
The conditional probability mass function of X given Y = y by
Math 425
Intro to Probability
Lecture 29
pX |Y