Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2006
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 3: Solutions
Due: March 1, 2006
1. The problem did not explicitly state that two cars canno
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 10
Topics: Poisson, Markov chains
Due: May 10th, 2006
1. All ships travel at the same speed
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 08
March 09, 2006
1. Random variables X and Y have the joint PDF shown below:
y
2.0
f (x,y)
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 09
March 21, 2006
1. Al and Bo are in a race. Denote Als and Bos elapsed times with random
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 11
April 4, 2006
1. A number p is drawn from the interval [0, 1] according to the uniform di
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 10
March 23, 2006
1. Suppose X is uniformly distributed between a and b.
a) Find the transfo
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 12
April 6, 2006
1. Widgets are packed into cartons which are packed into crates. The weight
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 18
May 2, 2006
1. (Example 5.15) Competing Exponentials. Two light bulbs have independent an
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 16
April 25, 2006
1. (Example 5.3) A computer executes two types of tasks, priority and non
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 19
May 4, 2006
1. (Example 6.3) A machine can be either working or broken down on a given da
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 21
Markov Chains: Absorption Probabilities and Expected Time to Absorption
May 11, 2006
1. J
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 14
April 11, 2006
1. Suppose four random variables, W , X, Y and Z, are known to be pairwise
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 20
Markov Chains: Steady State Behavior
May 09, 2006
1. (Problem 6.9) A professor gives test
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Tutorial 07
April 67, 2006
1. Suppose you are playing roulette with a biased wheel, such that your ch
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Tutorial 11
May 45, 2006
1. (Problem 5.14) Each morning as you pull out of your driveway you would lik
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 11:
Topic: Markov Processes
Due: May 12, 2006
1. At the Probability Coee House of MIT, ther
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 6
Due: April 5, 2006
1. Suppose that
MX (s) =
1
1
2
3
+
.
3 1s 3 3s
What is the PDF of X?
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2006
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 4: Solutions
Due: March 8, 2006
1. (a) Use the total probability theorem by conditioning on
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2006
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 5 Solutions
Due: March 22, 2006
1. We are given that F () =
1
2
ex dx =
0
0
1
2
ex dx = ex
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2006
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 6: Solutions
Due: April 5, 2006
1. X is the mixture of two exponential random variables wit
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2006
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 7: Solutions
Due: April 12, 2006
1. For both parts (a) and (b) we will make use of the form
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2006
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 9
Topics: Bernoulli, Poisson
Due: May 3rd, 2006
1. A successful call occurs with probabilit
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2006
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science 6.041/6.431: Probabilistic Systems Analysis (Spring 2006)
Problem Set 10 Topics: Poisson, Markov chains Due: May 10th, 2006 1. (a) We are given that the previous
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 1
Due: February 15, 2006
1. Express each of the following events in terms of the events A,
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 3
Due: March 1, 2006
1. Mary and Tom park their cars in an empty parking lot that consists
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 9
Topics: Bernoulli, Poisson
Due: May 3rd, 2006
1. Fred is giving out samples of dog food.
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Recitation 07
March 07, 2006
1. The random variable X is exponentially distributed with parameter .
fX
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 5
Due: March 22, 2006
1. Consider an exponentially distributed random variable X with param
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 4
Due: March 8, 2006
1. Professor May B. Right often has her science facts wrong, and answe
Probabilistic Systems Analysis and Applied Probability
EECS 6.041: A

Spring 2015
Massachusetts Institute of Technology
Department of Electrical Engineering & Computer Science
6.041/6.431: Probabilistic Systems Analysis
(Spring 2006)
Problem Set 7
Due: April 12, 2006
1. (a) Imagine that you rst roll a fair 6sided die and then you ip a