 11.1: Let X0, X1, X2,... be a Markov chain. Show that X0, X2, X4, X6,... ...
 11.2: Let X0, X1, X2,... be an irreducible Markov chain with state space ...
 11.3: A Markov chain has two states, A and B, with transitions as follows...
 11.4: Consider the Markov chain shown below, where 0 p< 1 and the labels ...
 11.5: Consider the Markov chain shown below, with state space {1, 2, 3, 4...
 11.6: Daenerys has three dragons: Drogon, Rhaegal, and Viserion. Each dra...
 11.7: A Markov chain X0, X1,... with state space {3, 2, 1, 0, 1, 2, 3} pr...
 11.8: Let G be an undirected network with nodes labeled 1, 2,...,M (edges...
 11.9: (a) Consider a Markov chain on the state space {1, 2,..., 7} with t...
 11.10: Let Xn be the price of a certain stock at the start of the nth day,...
 11.11: In chess, the king can move one square at a time in any direction (...
 11.12: A chess piece is wandering around on an otherwise vacant 8 8 chessb...
 11.13: Find the stationary distribution of the Markov chain shown below, w...
 11.14: There are two urns with a total of 2N distinguishable balls. Initia...
 11.15: Nausicaa Distribution sells distribution plushies on Etsy. They hav...
 11.16: This exercises considers random walk on a weighted undirected netwo...
 11.17: A cat and a mouse move independently back and forth between two roo...
 11.18: Let {Xn} be a Markov chain on states {0, 1, 2} with transition matr...
 11.19: Consider the following Markov chain on the state space {1, 2, 3, 4,...
 11.20: Let Q be the transition matrix of a Markov chain on the state space...
 11.21: In the game called Chutes and Ladders, players try to be first to r...
Solutions for Chapter 11: Markov Chains
Full solutions for Introduction to Probability  1st Edition
ISBN: 9781466575578
Solutions for Chapter 11: Markov Chains
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aerror (or arisk)
In hypothesis testing, an error incurred by failing to reject a null hypothesis when it is actually false (also called a type II error).

Acceptance region
In hypothesis testing, a region in the sample space of the test statistic such that if the test statistic falls within it, the null hypothesis cannot be rejected. This terminology is used because rejection of H0 is always a strong conclusion and acceptance of H0 is generally a weak conclusion

Adjusted R 2
A variation of the R 2 statistic that compensates for the number of parameters in a regression model. Essentially, the adjustment is a penalty for increasing the number of parameters in the model. Alias. In a fractional factorial experiment when certain factor effects cannot be estimated uniquely, they are said to be aliased.

Analysis of variance (ANOVA)
A method of decomposing the total variability in a set of observations, as measured by the sum of the squares of these observations from their average, into component sums of squares that are associated with speciic deined sources of variation

Binomial random variable
A discrete random variable that equals the number of successes in a ixed number of Bernoulli trials.

Causeandeffect diagram
A chart used to organize the various potential causes of a problem. Also called a ishbone diagram.

Comparative experiment
An experiment in which the treatments (experimental conditions) that are to be studied are included in the experiment. The data from the experiment are used to evaluate the treatments.

Completely randomized design (or experiment)
A type of experimental design in which the treatments or design factors are assigned to the experimental units in a random manner. In designed experiments, a completely randomized design results from running all of the treatment combinations in random order.

Continuous distribution
A probability distribution for a continuous random variable.

Convolution
A method to derive the probability density function of the sum of two independent random variables from an integral (or sum) of probability density (or mass) functions.

Covariance matrix
A square matrix that contains the variances and covariances among a set of random variables, say, X1 , X X 2 k , , … . The main diagonal elements of the matrix are the variances of the random variables and the offdiagonal elements are the covariances between Xi and Xj . Also called the variancecovariance matrix. When the random variables are standardized to have unit variances, the covariance matrix becomes the correlation matrix.

Critical value(s)
The value of a statistic corresponding to a stated signiicance level as determined from the sampling distribution. For example, if PZ z PZ ( )( .) . ? =? = 0 025 . 1 96 0 025, then z0 025 . = 1 9. 6 is the critical value of z at the 0.025 level of signiicance. Crossed factors. Another name for factors that are arranged in a factorial experiment.

Discrete distribution
A probability distribution for a discrete random variable

Event
A subset of a sample space.

Exhaustive
A property of a collection of events that indicates that their union equals the sample space.

Fisher’s least signiicant difference (LSD) method
A series of pairwise hypothesis tests of treatment means in an experiment to determine which means differ.

Forward selection
A method of variable selection in regression, where variables are inserted one at a time into the model until no other variables that contribute signiicantly to the model can be found.

Fraction defective control chart
See P chart

Frequency distribution
An arrangement of the frequencies of observations in a sample or population according to the values that the observations take on

Harmonic mean
The harmonic mean of a set of data values is the reciprocal of the arithmetic mean of the reciprocals of the data values; that is, h n x i n i = ? ? ? ? ? = ? ? 1 1 1 1 g .