- 4.1.1: The current in a certain circuit as measured by an ammeteris a cont...
- 4.1.2: Suppose the reaction temperature X (in 8C) in a certain chemical pr...
- 4.1.3: The error involved in making a certain measurement is acontinuous r...
- 4.1.4: Let X denote the vibratory stress (psi) on a wind turbine blade at ...
- 4.1.5: A college professor never finishes his lecture before theend of the...
- 4.1.6: The actual tracking weight of a stereo cartridge that is setto trac...
- 4.1.7: The article Second Moment Reliability Evaluationvs. Monte Carlo Sim...
- 4.1.8: In commuting to work, a professor must first get on a busnear her h...
- 4.1.9: Based on an analysis of sample data, the articlePedestrians Crossin...
- 4.1.10: A family of pdfs that has been used to approximate thedistribution ...
Solutions for Chapter 4.1: Probability Density Functions
Full solutions for Probability and Statistics for Engineering and the Sciences | 9th Edition
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
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
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.
Box plot (or box and whisker plot)
A graphical display of data in which the box contains the middle 50% of the data (the interquartile range) with the median dividing it, and the whiskers extend to the smallest and largest values (or some deined lower and upper limits).
An attribute control chart that plots the total number of defects per unit in a subgroup. Similar to a defects-per-unit or U chart.
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 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.
Defect concentration diagram
A quality tool that graphically shows the location of defects on a part or in a process.
Defects-per-unit control chart
See U chart
Discrete random variable
A random variable with a inite (or countably ininite) range.
The expected value of a random variable X is its long-term average or mean value. In the continuous case, the expected value of X is E X xf x dx ( ) = ?? ( ) ? ? where f ( ) x is the density function of the random variable X.
Exponential random variable
A series of tests in which changes are made to the system under study
Finite population correction factor
A term in the formula for the variance of a hypergeometric random variable.
A model that contains only irstorder terms. For example, the irst-order response surface model in two variables is y xx = + ?? ? ? 0 11 2 2 + + . A irst-order model is also called a main effects model
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.
Gamma random variable
A random variable that generalizes an Erlang random variable to noninteger values of the parameter r
Another name for the normal distribution, based on the strong connection of Karl F. Gauss to the normal distribution; often used in physics and electrical engineering applications
Geometric random variable
A discrete random variable that is the number of Bernoulli trials until a success occurs.
In multiple regression, the matrix H XXX X = ( ) ? ? -1 . This a projection matrix that maps the vector of observed response values into a vector of itted values by yˆ = = X X X X y Hy ( ) ? ? ?1 .