###### 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 ...

The problem should assume that a fraction q...
Question: 8.2.5

## Let X be a continuous random variable with values normally distributed over (−∞,+∞) ...

(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 ...

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 ...

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 ...

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 ...

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 ...

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. ...

Recall that m_{ij}= \sum\limits_{j}{\frac{z...
Question: 11.5.18