CSCI 1550
Randomized Algorithms and Probabilistic Analysis
1
Prof. Eli Upfal
Homework 2
DUE: 2/12/2015
Please Note:
Assignments will be turned in in-class. Assignments must be typeset in Latex. Answers must
be concise and correct. No late homework will be
Homework 5
CS 155 Spring 2013
March 15, 2013
Problem 1
(4.18)
have
Dene the random variables Xi = pi bi (t), 1 i n, and let X =
n
i=1 Xi .
We
1
E etXi = (etpi + etpi ).
2
x we have (using the same argument in Theorem 4.7)
By using the Taylor series expans
Problem 2.32
Part a
Let Ei be the event that the ith candidate is the best and he is hired.
P r(Ei ) = P r(candidate Ei is chosen|Ei is the best) P r(Ei is the best)
1
n
P r(candidate Ei is chosen|Ei is the best) =
P r(Ei is the best) =
P r(None of the rs
CS155 - Homework 1
Justin Oh(sjo)
Homework 1
Problem 1
(a) The pooled sample of k people will be positive if at least one of the k individuals is positive.
The probability that none of individuals are positive is (1 p)k . Thus the probability of the
poole
CS155
Probabilistic Methods in CS
Upfal
Homework 5
Due: Thu 5 March 2015, 2:30pm
Assignments will be handed in class. Assignments must be typeset in Latex.
Answers must be concise and correct. No late homework will be accepted
without prior authorization
CS155 - Homework 1
Lorenzo De Stefani(lorenzo)
Homework 1
Problem 1
(a) The probability that exactly k ips occur in traversing the n relays is given by:
n k
p (1 p)nk ,
k
(1)
for i = 0, 1, 2, . . . , n where n is the number of all possible conguration of
CS155: Probability and Computing:
Randomized Algorithms and Probabilistic
Analysis
Eli Upfal
Eli Upfal@brown.edu
Office: 319
TAs: Ahmad Mahmoody
and
Shashwat Silas
cs155tas@cs.brown.edu
It is remarkable that this science, which originated
in the considera
CS155 - Homework 3
()
Homework 3
Problem 1
(a) Note rst that:
t
t
E[(1/t)
t
Xi ] = (1/t)
i=1
E[Xi ] = (1/t)
i=1
E[X] = E[X]
i=1
and since the Xi s are independent:
t
t
Xi ) = (1/t)2 V (
V (1/t)
i=1
t
Xi ) = (1/t)2
i=1
t
V (Xi ) = (1/t)2
i=1
V (X) = V (X)/
The Probabilistic Method
Compute
n
i
i=0
n
i
1
2
n
The Probabilistic Method
Compute
n
n
i
i
i=0
n
1
2
Let X B(n.1/2),
n
E [X ] =
i
i=0
n
i
1
2
n
=
n
2
We prove a deterministic statement using a probabilistic
argument!
The Probabilistic Method
1
If E [
Random Variable
Denition
A random variable X on a sample space is a real-valued
function on ; that is, X : R. A discrete random variable is
a random variable that takes on only a nite or countably innite
number of values.
Discrete random variable X and re
CS155 - Homework 4
solution (solution)
Homework 4
Problem 1
(a) By tessellating the original square S we obtain n/ (b log n) sub-squares sj , with j cfw_1, 2, . . . , n/b log n
each of size b log n. Let us dene the following events:
1 : every sub-square h
CS155
Probabilistic Methods in CS
Upfal
Homework 6
Due: Thu 12 March 2015, 2:30pm
Assignments will be handed in class. Assignments must be typeset in Latex.
Answers must be concise and correct. No late homework will be accepted
without prior authorization
CS155
Probabilistic Methods in CS
Upfal
Homework 3
Due: Thu 19 Feb 2015, 2:30pm
Assignments will be handed in class. Assignments must be typeset in Latex.
Answers must be concise and correct. No late homework will be accepted
without prior authorization f
CSCI 1550
Randomized Algorithms and Probabilistic Analysis
1
Prof. Eli Upfal
Homework 1
DUE: 2/5/2015
Please Note:
Assignments will be handed in class. Assignments must be typeset in Latex. Answers must be
concise and correct. No late homework will be acc
CS155
Probabilistic Methods in CS
Upfal
Homework 4
Due: Thu 26 Feb 2015, 2:30pm
Assignments will be handed in class. Assignments must be typeset in Latex.
Answers must be concise and correct. No late homework will be accepted
without prior authorization f
CS155 - Homework 2
()
Homework 2
Problem 1
Let Yi be the result of the i-th spin. Since the probability of winning in a certain round is 1 , we
2
1
know that P r(Yi = 1) = 1 and P r(Yi = 0) = 2 . Let Y be the number of spins until you win. The
2
probabili