- 5-9.s5-13: A random variable X has the discrete uniform distribution (a) Show ...
- 5-9.s5-14: A random variable X has the Poisson distribution
- 5-9.s5-15: The geometric random variable X has probability distribution f 1x2 ...
- 5-9.s5-16: The chi-squared random variable with k degrees of freedom has momen...
- 5-9.s5-17: A continuous random variable X has the following probability distri...
- 5-9.s5-18: The continuous uniform random variable X has density function (a) S...
- 5-9.s5-19: A random variable X has the exponential distribution (a) Show that ...
- 5-9.s5-20: A random variable X has the gamma distribution
- 5-9.s5-21: Let X1, X2, . . . , Xr be independent exponential random variables ...
- 5-9.s5-22: Suppose that Xi has a normal distribution with mean i and variance ...
- 5-9.s5-23: Show that the moment generating function of the chi-squared random ...
- 5-9.s5-24: Continuation of Exercise S5-20. (a) Show that by expanding etX in a...
Solutions for Chapter 5-9: MOMENT GENERATING FUNCTIONS (CD ONLY)
Full solutions for Applied Statistics and Probability for Engineers | 3rd Edition
In a fractional factorial experiment when certain factor effects cannot be estimated uniquely, they are said to be aliased.
The arithmetic mean of a set of numbers x1 , x2 ,…, xn is their sum divided by the number of observations, or ( / )1 1 n xi t n ? = . The arithmetic mean is usually denoted by x , and is often called the average
Bivariate normal distribution
The joint distribution of two normal random variables
A horizontal line on a control chart at the value that estimates the mean of the statistic plotted on the chart. See Control chart.
The portion of the variability in a set of observations that is due to only random forces and which cannot be traced to speciic sources, such as operators, materials, or equipment. Also called a common cause.
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 probability distribution for a continuous random variable.
See Control chart.
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 .
Another name for factors that are arranged in a factorial experiment.
Defects-per-unit control chart
See U chart
An experiment in which the tests are planned in advance and the plans usually incorporate statistical models. See Experiment
A concept in parameter estimation that uses the variances of different estimators; essentially, an estimator is more eficient than another estimator if it has smaller variance. When estimators are biased, the concept requires modiication.
Erlang random variable
A continuous random variable that is the sum of a ixed number of independent, exponential random variables.
A property of a collection of events that indicates that their union equals the sample space.
Exponential random variable
A series of tests in which changes are made to the system under study
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.
The harmonic mean of a set of data values is the reciprocal of the arithmetic mean of the reciprocals of the data values; that is, h n x i n i = ? ? ? ? ? = ? ? 1 1 1 1 g .