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 E3658 HWK 3 - Conditional Probability, Gambler's Ruin (Solutions)
Fall 2013
Prepared by Charles Courtiol ([email protected])
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 8 - 2D CLT, Inequalities
(Solutions)
Fall 2013
Prepared by Charles Courtiol ([email protected])
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 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 5 - Normal RVs, CLT, Derived Distributions (Solutions)
Fall 2013
Prepared by Charles Courtiol ([email protected])
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
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 7 - Joint Distributions (continuous) (Solutions)
Fall 2013
Prepared by Charles Courtiol ([email protected])
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 #2
September 26, 2015
Page 1 of 2
Assignment #2 due September 30th, 2015
1. Stans birthday is approaching. He thinks that there is a 0.6 chance that his wife will get
him a new pair of red sus
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
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 #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 #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 #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
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
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
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
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
Homework 1 - Sets, Counting (Solutions)
Fall 2013
Prepared by Charles Courtiol ([email protected])
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 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)
IEOR 3658
Probability
Prof. Mariana Olvera-Cravioto
Assignment #6
October 19, 2011
Page 1 of 2
Assignment #6 due October 26th, 2011
1. A Cure for the Common Cold. The number of times that an individual contracts a
cold in a given year is a Poisson random
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(