550.420
Introduction to Probability
Exercises on Combinatorics
[1] A salad bar has 3 choices of greens, 8 veggies, 5 fruits, 3 dairy items, and 4 dressings. In how many
ways can I serve myself a salad
550.420
Introduction to Probability
Spring 2014
Midterm Examination # 1
Wednesday March 5, 2014
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NAME (Please print clearly):
SECTION:
I agree to complete this examination without unauthorized assist
550.420
Introduction to Probability
Properties of Independence
Subtlety
If P[E] = 0, then event E is independent of every
other event.
P[ EF ] P[ E ] 0 P[ EF ] 0
P[ E ] 0 P[ E ]P[ F ] 0
The condition
550.420
Introduction to Probability
Hypergeometric Distribution
Hypergeometric Distribution
Analogous to the Binomial distribution, but
models sampling without replacement.
Consider an urn model:
s ba
550.420
Introduction to Probability
Using Partial Information
Benefits of Conditional Probability
A framework for using partial knowledge or
incomplete information to revise probabilities.
A method fo
550.420
Introduction to Probability
Mode of the Binomial Distribution
What is the Most Likely Value?
What is the most likely value of the
Binomial(n,p) random variable?
i.e. which value of p(k) is the
550.420
Introduction to Probability
Multinomial Coefficients
Multinomial Coefficients
Divide n distinct elements into k distinct
subsets of sizes n1, n2, . , nk, where the ni
sum to n.
Alternatively,
550.420
Introduction to Probability
MORE ON INDEPENDENCE
Infinite Sets of Independent Events
Raise the definition one more level:
An infinite set of events are mutually
independent if every finite set
550.420
Introduction to Probability
FIRST STEP DECOMPOSITION
Which Occurs First?
Consider an infinite sequence of independent
repeated trials of a random experiment with
outcomes E and F among others.
550.420
Introduction to Probability
Geometric Distribution
Geometric(p) Distribution
Consider an infinite sequence of Bernoulli(p) trials.
On which trial does the first S occur?
Define X = # trial of
550.420
Introduction to Probability
Fall 2016
Midterm Examination # 1
Friday October 7, 2016
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NAME (Please print clearly):
SECTION:
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550.420
Introduction to Probability
Indicator Random Variables
Indicator Random Variable
I A ( ) 1
if A
0
if A
E[ I A ] 1P[ I A 1] 0 P[ I A 0]
1P[ A]
P[ A]
Usefulness
Indicators do book-keeping for u
550.420
Introduction to Probability
Independence and Mutual Independence
Revising Information
Consider events E and F.
There are three possible relationships:
F provides positive information about E:
550.420
Introduction to Probability
Gamma Function
Gamma Function
(t ) e y y t 1dy
0
e y y t 1
0
e y (t 1) y t 2 dy
0
(t 1) e y y t 2 dy
0
(t 1) (t 1)
Calculating the Gamma Function
Work backwards
550.420
Introduction to Probability
Discrete Random Variables
Discrete Random Variable
S is a countable sample space.
F is the power set of S
A discrete random variable X is a real-valued function
def
550.420
Introduction to Probability
Expectation of Discrete Random Variables
Expectation (Mean, Expected Value)
E[ X ]
k p
X
(k )
kR ( X )
R(X) denotes the range of X.
Properties of Expectation
What
550.420
Introduction to Probability
Uniform Distribution
Uniform(a,b) Distribution
Random value in the interval (a,b)
Density function
Distribution function
Slope = 1/(b-a)
1/(b-a)
a
b
a
b
Formulas
fU
550.420
Introduction to Probability
Spring 2014
Final Examination
9 AM Noon Wednesday May 14, 2014
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NAME (Please print clearly):
SECTION:
I agree to complete this examination without unauthorized ass
550.420
Introduction to Probability
Spring 2014
Midterm Examination # 2
Wednesday April 4, 2014
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NAME (Please print clearly):
SECTION:
I agree to complete this examination without unauthorized assist
550.420
Introduction to Probability
Spring 2014
Midterm Examination # 3
Wednesday April 30, 2014
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NAME (Please print clearly):
SECTION:
I agree to complete this examination without unauthorized assis
550.420
Introduction to Probability
Spring 2013
Midterm Examination # 3
Wednesday May 1, 2013
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NAME (Please print clearly):
SECTION:
I agree to complete this examination without unauthorized assistan
550.420
Introduction to Probability
Spring 2013
Midterm Examination # 1
Friday March 1, 2013
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NAME (Please print clearly):
SECTION:
I agree to complete this examination without unauthorized assistanc
550.420
Introduction to Probability
Spring 2013
Final Examination
2 5 PM Thursday May 16, 2013
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Course Goal
This course studies randomness and
uncertainty, with the intention of preparing
you to understand its occurrence and
consequences in your life and career.
The course provides the necessary
Registration Issues
There is really no class limit. If you are limited out, I
will sign you in if you have the prerequisites. But I
will try to keep the section sizes as equal as possible.
Graduate st
Homework 11
553.420/620 Introduction to Probability
Fall 2017
Due Thursday, November 16, 2017
Because of the exam last Monday, there are only two lectures discussing new material this week.
However, t
Homework 10
553.420/620 Introduction to Probability
Fall 2017
Due Thursday, November 9, 2017
[1] Two random points are selected from the interval (0, 1) independently. Find the probability that one
of
550.420
Introduction to Probability
Poisson Distribution
Rare Events in Bernoulli Trials
Suppose we want to model occurrence of a rare
event.
n trials
Success probability /n,
where is small or moderat
550.420
Introduction to Probability
Probabilists Have Their Moments
Moments
Notation:
E[X] =
k-th moment:
E[Xk]
k-th central moment
E[(X- )k]
Variance
Var(X) = 2(X) = E[(X-)2]
Variance is the average