- 7.3.1: Consider a system with two components [ASH 1970]. We observe the st...
- 7.3.2: Assume that a computer system is in one of three states: busy, idle...
- 7.3.3: Any transition probability matrix P is a stochastic matrix; that is...
- 7.3.4: Show that the Markov chain of Example 7.12 is irreducible and aperi...
Solutions for Chapter 7.3: State Classification And Limiting Probabilitites
Full solutions for Probability and Statistics with Reliability, Queuing, and Computer Science Applications | 2nd Edition
Additivity property of x 2
If two independent random variables X1 and X2 are distributed as chi-square with v1 and v2 degrees of freedom, respectively, Y = + X X 1 2 is a chi-square random variable with u = + v v 1 2 degrees of freedom. This generalizes to any number of independent chi-square random variables.
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.
A study in which a sample from a population is used to make inference to a future population. Stability needs to be assumed. See Enumerative study
The arithmetic mean of a set of numbers x1 , x2 ,…, xn is their sum divided by the number of observations, or ( / )1 1 n xi t n ? = . The arithmetic mean is usually denoted by x , and is often called the average
An estimator for a parameter obtained from a Bayesian method that uses a prior distribution for the parameter along with the conditional distribution of the data given the parameter to obtain the posterior distribution of the parameter. The estimator is obtained from the posterior distribution.
An effect that systematically distorts a statistical result or estimate, preventing it from representing the true quantity of interest.
The joint probability distribution of two random variables.
A chart used to organize the various potential causes of a problem. Also called a ishbone diagram.
A horizontal line on a control chart at the value that estimates the mean of the statistic plotted on the chart. See Control chart.
A linear function of treatment means with coeficients that total zero. A contrast is a summary of treatment means that is of interest in an experiment.
A graphical display used to monitor a process. It usually consists of a horizontal center line corresponding to the in-control value of the parameter that is being monitored and lower and upper control limits. The control limits are determined by statistical criteria and are not arbitrary, nor are they related to speciication limits. If sample points fall within the control limits, the process is said to be in-control, or free from assignable causes. Points beyond the control limits indicate an out-of-control process; that is, assignable causes are likely present. This signals the need to ind and remove the assignable causes.
The amount of variability exhibited by data
Error mean square
The error sum of squares divided by its number of degrees of freedom.
Estimate (or point estimate)
The numerical value of a point estimator.
A property of a collection of events that indicates that their union equals the sample space.
The distribution of the random variable deined as the ratio of two independent chi-square random variables, each divided by its number of degrees of freedom.
A type of experimental design in which every level of one factor is tested in combination with every level of another factor. In general, in a factorial experiment, all possible combinations of factor levels are tested.
Fixed factor (or fixed effect).
In analysis of variance, a factor or effect is considered ixed if all the levels of interest for that factor are included in the experiment. Conclusions are then valid about this set of levels only, although when the factor is quantitative, it is customary to it a model to the data for interpolating between these levels.
An arrangement of the frequencies of observations in a sample or population according to the values that the observations take on
Geometric random variable
A discrete random variable that is the number of Bernoulli trials until a success occurs.