- 4.6.1: Suppose that the pair(X, Y )is uniformly distributed on the interio...
- 4.6.2: Prove that if Var(X) < and Var(Y ) < , then Cov(X, Y ) is finite. H...
- 4.6.3: Suppose that X has the uniform distribution on the interval [2, 2] ...
- 4.6.4: Suppose that the distribution of a random variable X is symmetric w...
- 4.6.5: For all random variables X and Y and all constants a, b, c, and d, ...
- 4.6.6: Let X and Y be random variables such that 0 < 2 X < and 0 < 2 Y < ....
- 4.6.7: Let X, Y , and Z be three random variables such that Cov(X, Z) and ...
- 4.6.8: Suppose that X1,...,Xm and Y1,...,Yn are random variables such that...
- 4.6.9: Suppose that X and Y are two random variables, which may be depende...
- 4.6.10: Suppose that X and Y are negatively correlated. Is Var(X + Y ) larg...
- 4.6.11: Show that two random variables X and Y cannot possibly have the fol...
- 4.6.12: Suppose that X and Y have a continuous joint distribution for which...
- 4.6.13: Suppose that X and Y are random variables such that Var(X) = 9, Var...
- 4.6.14: Suppose that X, Y , and Z are three random variables such that Var(...
- 4.6.15: Suppose that X1,...,Xn are random variables such that the variance ...
- 4.6.16: Consider the investor in Example 4.2.3 on page 220. Suppose that th...
- 4.6.17: Let X and Y be random variables with finite variance. Prove that |(...
- 4.6.18: Let X and Y be random variables with finite variance. Prove that |(...
Solutions for Chapter 4.6: 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
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.
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.
A distribution with two modes
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 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.
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 measure of association between two random variables obtained as the expected value of the product of the two random variables around their means; that is, Cov(X Y, ) [( )( )] =? ? E X Y ? ? X Y .
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.
Degrees of freedom.
The number of independent comparisons that can be made among the elements of a sample. The term is analogous to the number of degrees of freedom for an object in a dynamic system, which is the number of independent coordinates required to determine the motion of the object.
Deming’s 14 points.
A management philosophy promoted by W. Edwards Deming that emphasizes the importance of change and quality
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.
The variance of an error term or component in a model.
A subset of a sample space.
Fraction defective control chart
See P chart
A function used in the probability density function of a gamma random variable that can be considered to extend factorials
Gamma random variable
A random variable that generalizes an Erlang random variable to noninteger values of the parameter r
A function that is used to determine properties of the probability distribution of a random variable. See Moment-generating function
Effects in a fractional factorial experiment that are used to construct the experimental tests used in the experiment. The generators also deine the aliases.