- 4.5.1: Prove that the 1/2 quantile as defined in Definition 3.3.2 is a med...
- 4.5.2: Suppose that a random variable X has a discrete distribution for wh...
- 4.5.3: Suppose that a random variable X has a continuous distribution for ...
- 4.5.4: . In a small community consisting of 153 families, the number of fa...
- 4.5.5: Suppose that an observed value of X is equally likely to come from ...
- 4.5.6: Suppose that a random variable X has a continuous distribution for ...
- 4.5.7: Suppose that a persons score X on a certain examination will be a n...
- 4.5.8: Suppose that the distribution of a random variable X is symmetric w...
- 4.5.9: Suppose that a fire can occur at any one of five points along a roa...
- 4.5.10: If n houses are located at various points along a straight road, at...
- 4.5.11: Let X be a random variable having the binomial distribution with pa...
- 4.5.12: Consider a coin for which the probability of obtaining a head on ea...
- 4.5.13: Suppose that the distribution of X is symmetric Suppose that the di...
- 4.5.14: Find the median of the Cauchy distribution defined in Example 4.1.8.
- 4.5.15: Let X be a random variable with c.d.f. F. Suppose that a 1/2.
- 4.5.16: Let X be a random variable. Suppose that there exists a number m su...
- 4.5.17: Let X be a random variable. Suppose that there exists a number m su...
- 4.5.18: Prove the following extension of Theorem 4.5.1. Let m be the p quan...
Solutions for Chapter 4.5: Expectation
Full solutions for Probability and Statistics | 4th 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
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.
Continuous random variable.
A random variable with an interval (either inite or ininite) of real numbers for its range.
A term used for the quantity ( / )( ) 1 1 2 n xi i n ? = that is subtracted from xi i n 2 ? =1 to give the corrected sum of squares deined as (/ ) ( ) 1 1 2 n xx i x i n ? = i ? . The correction factor can also be written as nx 2 .
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.
In hypothesis testing, this is the portion of the sample space of a test statistic that will lead to rejection of the null hypothesis.
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.
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).
Error mean square
The error sum of squares divided by its number of degrees of freedom.
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.
A property of a collection of events that indicates that their union equals the sample space.
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.
A series of tests in which changes are made to the system under study
Extra sum of squares method
A method used in regression analysis to conduct a hypothesis test for the additional contribution of one or more variables to a model.
A method of variable selection in regression, where variables are inserted one at a time into the model until no other variables that contribute signiicantly to the model can be found.
In statistical quality control, that portion of a number of units or the output of a process that is defective.
An arrangement of the frequencies of observations in a sample or population according to the values that the observations take on
A function that is used to determine properties of the probability distribution of a random variable. See Moment-generating function