- 11.2.1: Show that if X is a normal random variable with parameters (, 2), t...
- 11.2.2: Let X1, X2, . . . , Xn be independent geometric random variables ea...
- 11.2.3: Let X1, X2, . . . , Xn be n independent exponential random variable...
- 11.2.4: Using moment-generating functions, show that the sum of n independe...
- 11.2.5: Let X1, X2, . . . , Xn be n independent gamma random variables with...
- 11.2.6: The probability is 0.15 that a bottle of a certain soda is underfil...
- 11.2.7: Let X and Y be independent binomial random variables with parameter...
- 11.2.8: LetX, Y , andZ be three independent Poisson random variables with p...
- 11.2.9: Mr. Watkins is at a train station, waiting to make a phone call. Th...
- 11.2.10: Let X N (1, 2) and Y N (4, 7) be independent random variables. Find...
- 11.2.11: The distribution of the IQ of a randomly selected student from a ce...
- 11.2.12: Vicki owns two department stores. Delinquent charge accounts at sto...
- 11.2.13: Let the joint probability density function of X and Y be bivariate ...
- 11.2.14: Let X be the height of a man, and let Y be the height of his daught...
- 11.2.15: The capacity of an elevator is 2700 pounds. If the weight of a rand...
- 11.2.16: The distributions of the grades of the students of probability and ...
- 11.2.17: Suppose that car mufflers last random times that are normally distr...
- 11.2.18: An elevator can carry up to 3500 pounds. The manufacturer has inclu...
- 11.2.19: Let the joint probability mass function of X1, X2, . . . , Xr be mu...
- 11.2.20: Kim is at a train station, waiting to make a phone call. Two public...
Solutions for Chapter 11.2: Sums of Independent Random Variables
Full solutions for Fundamentals of Probability, with Stochastic Processes | 3rd Edition
a-error (or a-risk)
In hypothesis testing, an error incurred by failing to reject a null hypothesis when it is actually false (also called a type II error).
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
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.
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
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.
When y fx = ( ) and y is considered to be caused by x, x is sometimes called a causal variable
A horizontal line on a control chart at the value that estimates the mean of the statistic plotted on the chart. See Control 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.
The probability of an event given that the random experiment produces an outcome in another event.
Conditional probability density function
The probability density function of the conditional probability distribution of a continuous random variable.
When a factorial experiment is run in blocks and the blocks are too small to contain a complete replicate of the experiment, one can run a fraction of the replicate in each block, but this results in losing information on some effects. These effects are linked with or confounded with the blocks. In general, when two factors are varied such that their individual effects cannot be determined separately, their effects are said to be confounded.
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.
See Control chart.
Cumulative normal distribution function
The cumulative distribution of the standard normal distribution, often denoted as ?( ) x and tabulated in Appendix Table II.
An expression sometimes used for nonlinear regression models or polynomial regression models.
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 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
Gamma random variable
A random variable that generalizes an Erlang random variable to noninteger values of the parameter r