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Statistical Mechanics
Introduction to Probability - Solutions Manual
292 SOLVED PROBLEMS
Question: 11.2.21
(Roberts7) A city is divided into 3 areas 1, 2, and 3. It is estimated that amounts u1, u2, and u3 of pollution are emitted each day from these three areas. A fraction qij of the pollution from region i ends up the next day at region j. A fraction qi =1−Σj qij > 0 goes into the atmosphere and ...
Verified Answer:
The problem should assume that a fraction
q...
Question: 8.2.5
Let X be a continuous random variable with values normally distributed over (−∞,+∞) ...
Verified Answer:
(a) 1, 1/4, 1/9 (b) 1 vs. .3173, .25 vs. .0455, .1...
Question: 11.2.33
Consider a random walker who moves on the integers 0, 1, …, N, moving one step to ...
Verified Answer:
In each case Exercise 27 shows that f(i) = biNf(N)...
Question: 11.2.31
In Example 11.11, define f(i) to be the proportion of G genes in state i. Show that f is ...
Verified Answer:
You can easily check that the proportion of G’s in...
Question: 11.2.27
Consider an absorbing Markov chain with state space S. Let f be a function defined on S ...
Verified Answer:
Use the solution to Exercise 24 with w = f.
Question: 11.5.23
Assume that an ergodic Markov chain has states s1,s2,…,sk. Let Sj^(n) denote the number ...
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Assume that the chain is started in state si. Let ...
Question: 11.5.21
A highly simplified game of “Monopoly” is played on a board with four squares as shown in ...
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The transition matrix is
P= \begin{matrix} ...
Question: 11.5.19
Show that, for an ergodic Markov chain (see Theorem 11.16), Σ j mijwj = Σj zjj -1 = K. ...
Verified Answer:
Recall that
m_{ij}= \sum\limits_{j}{\frac{z...
Question: 11.5.18
A perpetual craps game goes on at Charley’s. Jones comes into Charley’s on an evening ...
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Form a Markov chain whose states are the possible ...
Question: 11.5.17
An ergodic Markov chain is started in equilibrium (i.e., with initial probability vector ...
Verified Answer:
We know that wZ = w. We also know that mki = (zii ...
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