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% Homework for the course "Discrete Modeling",
% Winter quarter, 1998, Anne Greenbaum.
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{\scriptsize Discrete Modeling, Winter 1998} \hfill
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Review for Midterm (Wed., Feb. 11)
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The midterm will cover:
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\item
Ch. 2, {\em except} Secs. 2.8, 2.11.3, 2.11.4, 2.16, 2.17.
\item
Notes on Basic Probability, Monte Carlo Methods, and Markov Chains.
\item
Ch. 3, {\em except} Secs. 3.6 and 3.7.
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To review:
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\item
Make sure you can do all of the assigned homework exercises.
Also do Section 3.5, \#7, 8, 10, 11, 17, 18.
\item
For extra practice, choose additional related problems in Chs. 2 and 3.
Many have answers in the back.
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Make sure you understand {\em expected value}, {\em variance},
the {\em law of large numbers}, and the {\em central limit theorem}.
A random variable $X_1$ takes on the value $2$ with probability $.4$
and $3$ with probability $.6$. What is the expected value and variance
of $X_1$?
There are $10000$ independent identically distributed random variables
$X_1 , \ldots , X_{10000}$, each with the distribution described above.
Define the random variable $A$ by
\[
A = \frac{1}{10000} \sum_{i=1}^{10000} X_i .
\]
What is the expected value and variance of $A$?
Sketch its probability density function, labeling the mean and standard
deviation. What is the expected value and variance of $4A$?
\item
Make sure you understand random walks and Markov chains.
A walker can go to any of four positions: $0$, $1$, $2$, or $3$. From
positions $1$ or $2$, he moves right one space with probability $1/2$,
he moves left one space with probability $1/4$, and he stays where he
is with probability $1/4$. From position $0$, he moves right with
probability $1/2$ and stays where he is with probability $1/2$.
If he reaches position $3$, he stays there. He starts in position $1$.
What are his possible positions after $2$ steps and what is the probability
of each?
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