15-359/659: Probability and Computing
Jakub Pachocki
1
Recitation 2: Conditional Probability
January 23, 2015
Can YOU be a weather forecaster?
Exercise 1.1. In the hope of having a dry outdoor wedding, John and Mary decide to get married in the
desert, wh

15-359/659: Harchol-Balter
1. Evaluate:
Solution:
WARMUP 2: Calc
2y
dy
0 e
2y
e
dy =
0
e2y
2
y=
y=0
1
= 0 ( )
2
1
=
2
2. Evaluate:
Solution:
y
0 ye dy
You will need integration by parts.
Let u = y, and du = dy. Let dv = ey , and v = ey .
udv = uv
vdu
ye

1
Dimensionality reduction
Suppose we have a collection of n articles. We would like to eciently (approximately) answer
queries of the form: given two articles A and B, how many words are there that occur in exactly
one of A, B? We can aord some preproces

15-359/659: Probability and Computing
David Wajc
1
Lecture: Hashing and its Uses
March 18, 2015
Hashing
In many problems we are interested in computing properties of sets of elements from some universe U
(think of U as being all numeric or string identier

15-359/659 Probability and Computing
(SPRING 15), 12 Units
www.cs.cmu.edu/~15359/
CLASS: Mon/Wed 1:30 p.m. - 2:50 p.m. in Scaife Hall
125
RECITATION A: Fri 1:30 p.m. - 2:20 p.m. in Scaife
Hall 125
RECITATION B: Fri 2:30 p.m. - 3:20 p.m. in Scaife
Hall 125

15-359/15-659: Homework #8: Randomized Algorithms
Refer to Mors Randomized Algorithms Notes, 2013
Due: Fri, Mar 20, 2015, in recitation.
Instructions
Unless stated otherwise, you may not use Mathematica or any other computer algebra system to solve
homewo

More Hints on HW 9
1
Problem 2(d)
You dont need to actually use part (c) for this. Youve seen this analysis on a previous
homework, so it should be easy.
2
Problem 3
There are two things that are confusing people on this problem:
1. The rst is getting the

15359/659 Probability and Computing, Spring 2015
Homework 9
Due March 27 in Recitation.
Please submit Questions 1-3 and 4-6 separately.
1
Shuing to a Particular Song
You are playing music on your antique iPod. You have a particular song
that you would lik

21-127: Number Theory Notes Spring 2014
March 20, 2014
1
Motivation
For the next 8 lectures or so, we will be studying the theory of numbers. Since youve
been playing with numbers since you were very young, this topic should feel very familiar to
you. How

15-359/659: Harchol-Balter
1/12/15
WARMUP 4: Counting (Combinatorics)
1. Baskin Robins has n avors of ice cream. You are building a cone
with k < n scoops. How many dierent cones can you make if
each avor can only be used once, and the ordering of the avo

15-359/659: Harchol-Balter
DTMCs
1
4/1/15
Limiting Distribution vs. Stationary Distribution
1. Fill in denition of limiting distribution .
2. Fill in denition of stationary distribution .
2
Stationary distribution = Limiting distribution
Theorem 1 Given a

15-359/659: Harchol-Balter
Expectation and Variance
Mean of Geometric(p)
X Geometric(p)
1. Express E [X] as an innite sum:
2. Compute E [X] by conditioning instead of the innite sum.
1/26/15
Linearity of Expectations
Theorem: Let X and Y be random variabl

15-359/659: Probability and Computing
David Wajc
1
Recitation 8: Random Graphs
March 20, 2015
Random graphs
Consider constructing a random undirected graph on n vertices in the following manner. For each of
the n pairs of vertices, independently add an ed

15-359/659: Probability and Computing
Jakub Pachocki
1
Recitation 2: Conditional Probability
January 23, 2015
Can YOU be a weather forecaster?
Exercise 1.1. In the hope of having a dry outdoor wedding, John and Mary decide to get married in the
desert, wh

15-359/659: Probability & Computing
Name:
Midterm II
Problem #
Max Points
1
20
2
15
3
20
4
15
5
15
6
15
Total
Points Received
100
Please show your work throughout, and be careful to manage your time. Remember that answers are short!
THEOREMS FOR YOUR USE:

15-359/15-659: Homework #7: Chernoff Bounds
Refer to Mors Chernoff Bounds Notes, 2013
Due: Thurs, Mar 5, 2015, by midnight.
Instructions
Unless stated otherwise, you may not use Mathematica or any other computer algebra system to solve
homework problems.

Name:
Andrew ID:
15-359/15-659: Probability and Computing
Quiz #1: Conditional Expectation: Jan 26, 2015
Below is the joint pmf for random variables X and Y .
Derive E [X | Y 1].
X=1
X=2
0
0.2
0.5
0.1
Y=0 Y=1
Table 1: pX,Y (x, y)
1
0.2
0
Y=2

15-359/15-659: Homework #10: Discrete-time Markov Chains
Textbook: Performance Modeling & Design of Computer Systems, 2013
Due: Fri, April 3, 2015, in recitation.
Instructions
For Part III of the class, you are free to use Matlab or Mathematica to solve s