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First Course in Probability 8th Edition - Solutions by Chapter
Full solutions for First Course in Probability | 8th 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.
In a fractional factorial experiment when certain factor effects cannot be estimated uniquely, they are said to be aliased.
See Arithmetic mean.
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.
Axioms of probability
A set of rules that probabilities deined on a sample space must follow. See Probability
Central composite design (CCD)
A second-order response surface design in k variables consisting of a two-level factorial, 2k axial runs, and one or more center points. The two-level factorial portion of a CCD can be a fractional factorial design when k is large. The CCD is the most widely used design for itting a second-order model.
Central limit theorem
The simplest form of the central limit theorem states that the sum of n independently distributed random variables will tend to be normally distributed as n becomes large. It is a necessary and suficient condition that none of the variances of the individual random variables are large in comparison to their sum. There are more general forms of the central theorem that allow ininite variances and correlated random variables, and there is a multivariate version of the theorem.
Components of variance
The individual components of the total variance that are attributable to speciic sources. This usually refers to the individual variance components arising from a random or mixed model analysis of variance.
The mean of the conditional probability distribution of a random variable.
Conditional probability density function
The probability density function of the conditional probability distribution of a continuous random variable.
An estimator that converges in probability to the true value of the estimated parameter as the sample size increases.
Continuous random variable.
A random variable with an interval (either inite or ininite) of real numbers for its range.
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.
A dimensionless measure of the linear association between two variables, usually lying in the interval from ?1 to +1, with zero indicating the absence of correlation (but not necessarily the independence of the two variables).
In hypothesis testing, this is the portion of the sample space of a test statistic that will lead to rejection of the null hypothesis.
Defects-per-unit control chart
See U chart
A model to relate a response to one or more regressors or factors that is developed from data obtained from the system.
A study in which a sample from a population is used to make inference to the population. See Analytic study
Fisher’s least signiicant difference (LSD) method
A series of pair-wise hypothesis tests of treatment means in an experiment to determine which means differ.
An arrangement of the frequencies of observations in a sample or population according to the values that the observations take on