IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Lecture 24
December 6, 2011
Page 1 of 1
Lecture 24
1. Charity University announces a fund raising target of $10 millions. It sends out n letters
to its alumni. From past experience, there is a probabilit
IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Practice Second Midterm
November 7, 2011
Page 1 of 7
Practice Second Midterm
Place all answers on the question sheet provided. The exam is open textbook (Bertsekas and Tsitsiklis)
and open notes/handouts
IEOR E3658
HWK 8 - 2D CLT, Inequalities
(Solutions)
Fall 2013
Prepared by Charles Courtiol (cc3591@columbia.edu)
Exercise 1.
a To compute the marginal distribution we integrate the joint density:
+
s, fX (s) =
fX,Y (s, t)dt
1
2 s2 +4
e 2
=
+
e2t dt
1
2
2
IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Practice Final
December 6, 2011
Page 1 of 17
Practice Final
This Practice Final Exam is intended to be completed in 3 hours and it is worth a
total of 180 points. This examination consists of 17 printed
IEOR E3658 HWK 3 - Conditional Probability, Gambler's Ruin (Solutions)
Fall 2013
Prepared by Charles Courtiol (cc3591@columbia.edu)
Exercise 1.
PN =
N
N
a Trivially, P0 = 0 and PN = 1. Letting Pn =
= 1. Thus, boundary conditions are veried.
n
N
we have P0
IEOR E3658
HWK 7 - Joint Distributions (continuous) (Solutions)
Fall 2013
Prepared by Charles Courtiol (cc3591@columbia.edu)
a The set C = cfw_(X, Y ) : X 2 + Y 2 r2 represents the circle of center (0, 0)
and radius r such that the area enclosed by this
IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Assignment #4
October 4, 2011
Page 1 of 2
Assignment #4 due October 12th, 2008
1. The annual premium of a special kind of insurance starts at $1000 and is reduced by 10%
after each year where no claim ha
IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Assignment #10
November 17, 2011
Page 1 of 1
Assignment #10 due November 23rd, 2011
1. (From text) If X is a random variable that is uniformly distributed between -1 and 1, nd
the PDF of |X | and the PDF
IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Assignment #3
September 27, 2011
Page 1 of 2
Assignment #3 due October 5th, 2011
1. (From text) We deal from a well-shued 52-card deck. Calculate the probability that the
13th card is the rst king to be
IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Assignment #7
October 26, 2011
Page 1 of 2
Assignment #7 due November 2nd, 2011
1. Consider a random variable X that takes values in the interval (0, a) whose density function
has a triangular shape and
IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Assignment #8
November 2, 2011
Page 1 of 1
Assignment #8 due November 9th, 2011
1. The time Andrew will be at the dentist has an exponential distribution with mean of one hour.
He comes to a parking mete
IEOR E3658
Practice Midterm 2
Probability: Practice Midterm 2
Fall 2013
November, 2013
Solutions
UNI:
Name:
Instructions
Please write your answers on the question sheets. Answers written on scrap paper will not
be evaluated. Show and explain your work. Po
IEOR E3658
Midterm 2
Fall 2013
Probability: Midterm 2 Solutions
November, 2013
1. (30 points) Let U U nif (0, 1). We dene a new random variable Y by
Y = ln (1 U )
(a) (10 points) What is the range of the random variable Y ? In other words, what values
can
IEOR E3658
Practice Final - Solution
Fall 2013
Remark 1. Notice that these are suggested solutions. There are more than one (correct) way to
reach to the solution.
1. A class has N slots available, which will be lled on a First-Come-First-Serve basis. Eac
IEOR E3658
Final Solutions
Fall 2013
1. (20 points) Maria has invited 20 of her friends to a dinner party. She plans to serve three
dierent menus consisting of: sh, chicken, and beef. About 50% of the population prefers
beef, 20% chicken, and 30% sh. In a
IEOR E3658
Midterm 1 - Solutions
Fall 2013
Probability: Midterm 1 - Solutions
1. Three dierent machines M1 , M2 and M3 were used for producing a large batch of staplers.
Suppose that 20% of the staplers were produced by machine M1 , 30% by machine M2 and
IEOR E3658
Practice Midterm 1 - Solutions
Fall 2013
Probability: Practice Midterm 1 - Solutions
October, 2013
Name:
UNI:
Instructions
Please write your answers on the question sheets. Answers written on scrap paper will not
be evaluated. Show and explain
IEOR E3658
Homework 1 - Sets, Counting (Solutions)
Fall 2013
Prepared by Charles Courtiol (cc3591@columbia.edu)
Exercise 1.
For this exercise it is important to remember that:
Intersection () stands for "AND"
Union () stands for "OR"
a "Only F occurs" m
IEOR E3658 HWK 5 - Normal RVs, CLT, Derived Distributions (Solutions)
Fall 2013
Prepared by Charles Courtiol (cc3591@columbia.edu)
Exercise 1.
Let X1 , ., Xn independent random variables s.t. Xi N (i , i2 ).
a Using independence of the Xi s we will calcul
IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Assignment #3
September 30, 2015
Page 1 of ?
Assignment #3 due October 7th, 2015
1. We deal from a well-shued 52-card deck. Calculate the probability that the 13th card is the
rst king to be dealt.
2. Ei
Probabilistic Model:
1) sample space all possible outcomes of an experiment
2) probability law assigns to each set A a number P(A) is between [0,1] that encodes the knowledge or belief about the
collective likelihood of the elements of A
Definition: P(A|B
Probability Class 7 (9/30)
Example:
Suppose we want to count all possible anagrams of the word MISSISSIPPI
We permute all 11 letters assuming theyre all different, M I1 S1 S2 I2 S3 S4 I3 P1 P2 I4
There are 11! possible permutations
-> 11! / (4! (Ss) 4!(Is
Probability Recitation Examples
1. Independent trials consisting of rolling a pair of fair dice are performed. Calculate the probability that an
outcome of 5 appears before an outcome of 7 when outcome is defined as the sum of the pair of dice.
En : no 5
Probability Class 3
Example: Toss 2 fair 6 sided dice
What are the possible outcomes for the sum?
= cfw_1, 2, 3, 12
Compute the probability that sum is 5
(Approach for two dice can be to just draw a table of all combinations which gives 10 * 1/36 = 5/18)
Probability Class 2
How to describe a probabilistic model
Probabilistic Model:
1) sample space all possible outcomes of an experiment
2) probability law assigns to each set A a number P(A) is between [0,1] that encodes the knowledge or
belief about the co
Probability Class 4
Bayes Rule:
Let A1, A2, , An be disjoint events that form a partition of the , and suppose that P(Ai) > 0 for all i,
For any event B such that P(B) > 0, we have
P(Ai|B) = P(Ai B) / P(B) = P(B|Ai)P(Ai)/(P(B|A1)P(A1) + P(B|A2)+ P(B|An)P(
Probability Class 6 (9/28)
Example:
We want to count the number of UNIs possible. Each UNI consists of either 2 or 3 letters followed by
4 numbers between 0 and 4
2 letter + 4 numbers: _26_ _26_ _10_ _10_ _10_ _10_
3 letter + 4 numbers: _26_ _26_ _26_ _10
Probability Class 5
*
Next quiz cheat sheet allowed
- quiz only covers material up to the Wednesday that the HW was assigned
*
If A and B are independent events
P(A BC) = P(A) P(A B) = (because A and B are independent) P(A) P(A)P(B)
= P(A)(1-P(B) = P(A)P(
IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Assignment #1
September 14, 2015
Page 1 of 2
Assignment #1 due September 23rd, 2015
1. Let A and B be two sets.
(a) Show using Venn diagrams that
Ac = (Ac B) (Ac B c ),
B c = (A B c ) (Ac B c )
(b) Show