Probability Practices
Ex.
Let A and B be two disjoint events. Can you write IA + IB as a single indicator function?
Since A and B are disjoint,
for any w A B, IA(w) + IB(w) = 1
for any w / A B,
IA(w) + IB(w) = 0. Hence, IA + IB = IAB.
Ex.
Let A and B be t
Problem Set 10
36-217, S16
This homework is due Friday, April 22, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a
Problem Set 3
36-217, S16
This homework is due Friday, Feb. 5, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a fu
Problem Set 4
36-217, S16
This homework is due Friday, Feb. 12, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a f
Problem Set 4
36-217, S16
This homework is due Friday, Feb. 12, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a f
Problem Set 9
36-217, S16
This homework is due Friday, April 8, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a f
Problem Set 6
36-217, S16
This homework is due Thursday, March 3, at 5 PM. Late submissions are allowed until Monday, March
14, at 11AM. As a reminder, a separate paper submission is required for each problem. Failure to submit
a correctly formatted assig
Problem Set 7
36-217, S16
This homework is due Friday, Mar. 18, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a f
36-217 - Probability Theory and Random Processes
Fall 2013
Final Exam
December 16, 2013
Your ANDREW ID:
Your First and Last Name:
Your Signature:
Write the answers to the exam problems on blue books.
Make sure to write your name also on the blue book yo
Problem Set 8
36-217, S16
This homework is due Friday, March 25, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a
36-217 - Probability Theory and Random Processes
First Midterm
September, 26 2013
Your First and Last Name:
YOUR ANDREW ID:
Write the answers to the exam problems on a blue book.
Make sure to write your name and andrew ID also on the blue book you turn
Problem Set 2
36-217, S16
This homework is due Friday, Jan. 29, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a f
Problem Set 1
36-217, S16
This homework is due Friday, Jan. 22, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a f
Problem Set 5
36-217, S16
This homework is due Friday, Feb. 26, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a f
Probability Proofs
-For every events A and B, P(A B) = P(A) + P(B) P(A B).
Proof:
Ac B and A B are disjoint and (Ac B) (A B) = B.
so
P(B) = P(Ac B) + P(A B)
P(Ac B) = P(B) P(A B)
Also A and Ac B are disjoint and A (Ac B) = A B.
P(A B) = P(A) + P(Ac B)
so
Random Variables
-Random variables are functions from to R and are used to represent uncertainty about numbers.
-A random variable X is a function such that X : R.
-The indicator function is a special type of random variable. Consider an event A F. The in
Set Operations
-Let A and B be two sets. o is an element of the union between A and B, A B, if and only if
either
o is an element of A or o is an element of B. That is, A B = cfw_o : o A or o B.
-Let (An)nN be a sequence of sets. o is an element of the un
Sets
-A set is a collection of objects.
-If a set has a nite number of objects, o1, o2, . . . , on, we denote it by cfw_o1, o2, . . . , on.
-set of integers=Z, set of natural numbers=N, set of real numbers=R
Ex.
set of possible outcomes of a six sided die
Probability Axioms
-denote by the sample space, that is, the set of all possible outcomes in the situation we are interested
in.
- A probability is a function dened on subsets of which is usually interpreted in one the following
ways:
-(Frequency) P(A) de
Elementary Probability Examples
Ex.
Show that P(A Bc) (Ac B) = P(A) + P(B) 2P(A B)
A B = (A B)(A Bc)(Ac B)
Since the sets on the right hand side are disjoint
P(A B) = P(A B) + P(A Bc) + P(Ac B)
Also
P(A B) = P(A) + P(B) P(A B)
Together,
P(A B) + P(A Bc) +
Conditional Probability
-Let A and B be two events, the conditional probability of A given B is denoted by P(A|B).
- the uncertainty about A assuming that B is true.
-P(A B) = P(A)P(B|A)
-the probability of A and B happening is the same as the probability
Equiprobable Probability
-in problems with equiprobable outcomes, a probability of a set is proportional to its size.
-Consider you have n dierent objects. The number of ways you can select k times among them with
replacement is n^k.
Ex.
Consider a six si
Conditional Probability Exercises
Ex.
Let A and B be two events such that P(A) = 1/2, P(B) = 1/3 and P(A B) = 1/4.
P(A|B) = P(A|B) = P(A B)/P(B)
=1/4/1/3=3/4
P(A|Bc) = P(A Bc)/P(Bc)
=P(A) P(A B)/1 P(B)
=1/2 /2/3=3/8
P(B|A) = P(B A)/P(A)
=1/4/1/2=1/2
P(Ac|
Review Session
Taylor Pospisil, Richard Wang
Midterm 2
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Random Variables
Definition
A random variable is a function from the outcomes of the
experiment to the real numbers.
Exa
Problem Set 11
36-217, S16
This homework is due Friday, April 29, at 11 AM. As a reminder, a separate paper submission is required
for each problem. Failure to submit a correctly formatted assignment will result in partial credit for the
assignment. For a
a Our final is scheduled for Tuesday, 5/3, from ‘l—4PM
0 Please bring your student ED.
6 The exam will be comprehensive
0 You are expected to know or be able to figure out
the properties of the following distributions:
Binomial, Poisson, Geometric, Expon