Solutions for Chapter 3: Discrete Random Variables and Their Probability Distributions

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.

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.

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.

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.

An equation for a conditional probability such as PA B ( | ) in terms of the reverse conditional probability PB A ( | ).

Unbiased estimator.

The joint probability distribution of two random variables.

In experimental design, a group of experimental units or material that is relatively homogeneous. The purpose of dividing experimental units into blocks is to produce an experimental design wherein variability within blocks is smaller than variability between blocks. This allows the factors of interest to be compared in an environment that has less variability than in an unblocked experiment.

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

A subset selected without replacement from a set used to determine the number of outcomes in events and sample spaces.

The probability of an event given that the random experiment produces an outcome in another event.

The probability mass function of the conditional probability distribution of a discrete random variable.

A correction factor used to improve the approximation to binomial probabilities from a normal distribution.

A continuous random variable with range of a inite interval and a constant probability density function.

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 off-diagonal elements are the covariances between Xi and Xj . Also called the variance-covariance matrix. When the random variables are standardized to have unit variances, the covariance matrix becomes the correlation matrix.

For a random variable X, the function of X deined as PX x ( ) ? that is used to specify the probability distribution.

A management philosophy promoted by W. Edwards Deming that emphasizes the importance of change and quality

An experiment in which the tests are planned in advance and the plans usually incorporate statistical models. See Experiment

See P chart

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 .