Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 9 Solutions
1. (a) Yes, to 0. Applying the weak law of large numbers, we have
P(|Ui | > ) 0 a
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Recitation 20 Solutions: November 18, 2010
1. (a) Let Xi be a random variable indicating the quality of t
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Recitation 14 Solutions
October 26, 2010
1. (a) Let X = (time between successive mosquito bites) = (time
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 11
Never Due
Covered on Final Exam
1. Problem 7, page 509 in textbook
Derive the ML estimator
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 9
Due November 22, 2010
1. Random variable X is uniformly distributed between 1.0 and 1.0. Le
Massachusetts nstitute of Technology
Department of Electrical Engineering Computer Science
6 041/6 431: Probabilistic Systems Analysis
(Fall 2010)
6.041/6.431 Fall 2010 Final Exam Solutions
Wednesday December 15 9:00AM - 12:00noon.
Problem 1. 32 points) C
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 11 Solutions
1. Check book solutions.
2. (a) To nd the MAP estimate, we need to nd the value
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 7: Solutions
1. (a) The event of the ith success occuring before the jth failure is equivalen
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 10 Solutions
1. A nancial parable.
(a) The bank becomes insolvent if the assets gain R 5 (i.e
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Recitation 19 Solutions: November 16, 2010
1. (a) The Markov chain is shown below.
1
1
15
9
1/2
1/8
1/8
1
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Recitation 15 Solutions
October 28, 2010
1. (a) Let X be the time until the rst bulb failure. Let A (resp
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Recitation 17 Solutions
November 4, 2010
1. (a) K has a Poisson distribution with average arrival time =
Massachusetts nstitute of Technology
Department of Electrical Engineering Computer Science
6 041/6 431: Probabilistic Systems Analysis
(Fall 2009)
Quiz 2 Solutions:
November 3 2009
Problem 2. (49 points)
(a) (7 points)
We start by recognizing that fX (x)
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 7
Due November 8, 2010
1. Consider a sequence of mutually independent, identically distribute
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 10
Due December 2, 2010 (in recitation)
1. A nancial parable. An investment bank is managing
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Recitation 24: Solutions
December 7, 2010
1. (a) Normalization of the distribution requires:
1=
pK (k; )
Massachusetts nstitute of Technology
Department of Electrical Engineering Computer Science
6 041/6 431: Probabilistic Systems Analysis
(Fall 2010)
6 041/6 431 Fall 2010 Quiz 2 Solutions
Problem 1. 80 points) In this problem:
(i) X is a (continuous) unifor
Massachusetts nstitute of Technology
Department of Electrical Engineering Computer Science
6 041/6 431: Probabilistic Systems Analysis
(Fall 2009)
Final Solutions:
December 15 2009
Problem 2. (20 points)
(a) (5 points)
Were given that the joint PDF is con
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 8
Due November 15, 2010
1. Oscar goes for a run each morning. When he leaves his house for hi
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Problem Set 8: Solutions
1. (a) We consider a Markov chain with states 0, 1, 2, 3, 4, 5, where state i in
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Recitation 18: Solutions
November 9, 2010
1. a) The number of remaining green sh at time n completely det
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2010)
Recitation 21 Solutions
November 23, 2010
1. (a) To use the Markov inequality, let X =
10
i=1 Xi .
Then,
Massachusetts nstitute of Technology
Department of Electrical Engineering Computer Science
6 041/6 431: Probabilistic Systems Analysis
(Fall 2010)
Recitation 16 Solutions
6.041/6.431 Spring 2007 Quiz 2 Solutions)
November 2, 2010
Problem 1:
(a)
(i) The pl
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Fall 2009)
Quiz 1 Solutions:
October 13, 2009
1. (10 points) We start rst by listing the following probabilities:
P(