George Washington University
School of Engineering and Applied Science
Department of Engineering Management and Systems Engineering
EMSE 208 Stochastic Foundations Of Operations Research
Midterm Examination
Instructions: Answer every question of this shee

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Lecture 1: Introduction to
Probability Theory
Professor Thomas A. Mazzuchi
2
Overview of
Probability Terminology
We Model The Outcomes of Random Experiments
Outcome are unknown in advance
List of possible outcomes is known
Examples
Flipping a coin

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Lecture 2: Random Variables
Professor Thomas A. Mazzuchi
2
Random Variable
Definition
A random variable X, is a real valued function on a
probability space X:S such that for all t in we can
assign a probability to the event X t.
We use capital letters

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Markov Chains
Definition
A Stochastic Process cfw_X(t), tT is a collection of random
variables. That is, for each tT, Xcfw_t) is a random variable.
The values of t is often interpreted as time (though not
necessarily) and thus we say that X(t) is the

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Markov Chains
Time to revisit a recurrent state
j are often called stationary probabilities if we define
m j , j = E[# transitions to get j| the MC starts in j]
= 1/ j
Patterns
Consider a MC with transitions Pi,j and steady state probabilities
pj, st