- 8.1.1: Show that the solution to the matrix equation (8.17) with the initi...
- 8.1.2: Show that the solution to the matrixvector equation (8.18) can be w...
- 8.1.3: Show that the solution to the matrixvector equation (8.19) can be w...
- 8.1.4: Show that the solution to equation (8.14) for a nonhomogeneous CTMC...
- 8.1.5: For a homogeneous CTMC show that the Laplace transform of the trans...
- 8.1.6: Show that the integral (convolution) form of the Kolmogorov forward...
- 8.1.7: Show that 0 = 0 is an eigenvalue of the generator matrix Q.
Solutions for Chapter 8.1: Introduction
Full solutions for Probability and Statistics with Reliability, Queuing, and Computer Science Applications | 2nd Edition
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 portion of the variability in a set of observations that can be traced to speciic causes, such as operators, materials, or equipment. Also called a special cause.
Asymptotic relative eficiency (ARE)
Used to compare hypothesis tests. The ARE of one test relative to another is the limiting ratio of the sample sizes necessary to obtain identical error probabilities for the two procedures.
Average run length, or ARL
The average number of samples taken in a process monitoring or inspection scheme until the scheme signals that the process is operating at a level different from the level in which it began.
Coeficient of determination
See R 2 .
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.
The probability 1?a associated with a conidence interval expressing the probability that the stated interval will contain the true parameter value.
If it is possible to write a probability statement of the form PL U ( ) ? ? ? ? = ?1 where L and U are functions of only the sample data and ? is a parameter, then the interval between L and U is called a conidence interval (or a 100 1( )% ? ? conidence interval). The interpretation is that a statement that the parameter ? lies in this interval will be true 100 1( )% ? ? of the times that such a statement is made
Continuous uniform random variable
A continuous random variable with range of a inite interval and a constant probability density function.
In the most general usage, a measure of the interdependence among data. The concept may include more than two variables. The term is most commonly used in a narrow sense to express the relationship between quantitative variables or ranks.
Formulas used to determine the number of elements in sample spaces and events.
Cumulative normal distribution function
The cumulative distribution of the standard normal distribution, often denoted as ?( ) x and tabulated in Appendix Table II.
An experiment in which the tests are planned in advance and the plans usually incorporate statistical models. See Experiment
The amount of variability exhibited by data
Distribution free method(s)
Any method of inference (hypothesis testing or conidence interval construction) that does not depend on the form of the underlying distribution of the observations. Sometimes called nonparametric method(s).
An analysis of how the variance of the random variable that represents that output of a system depends on the variances of the inputs. A formula exists when the output is a linear function of the inputs and the formula is simpliied if the inputs are assumed to be independent.
Error sum of squares
In analysis of variance, this is the portion of total variability that is due to the random component in the data. It is usually based on replication of observations at certain treatment combinations in the experiment. It is sometimes called the residual sum of squares, although this is really a better term to use only when the sum of squares is based on the remnants of a model-itting process and not on replication.
A function used in the probability density function of a gamma random variable that can be considered to extend factorials
The geometric mean of a set of n positive data values is the nth root of the product of the data values; that is, g x i n i n = ( ) = / w 1 1 .
Goodness of fit
In general, the agreement of a set of observed values and a set of theoretical values that depend on some hypothesis. The term is often used in itting a theoretical distribution to a set of observations.