- 8.4.1: Let X and Y be independent random numbers from the interval (0, 1)....
- 8.4.2: Let X and Y be two positive independent continuous random variables...
- 8.4.3: Let X N (0, 1) and Y N (0, 1) be independent random variables. Find...
- 8.4.4: From the interval (0, 1), two random numbers are selected independe...
- 8.4.5: Let 1/9 < c < 1/9 be a constant. Let p(x, y), the joint probability...
- 8.4.6: Let X and Y be independent random variables with common probability...
- 8.4.7: Let X and Y be independent random variables with common probability...
- 8.4.8: Prove that if X and Y are independent standard normal random variab...
- 8.4.9: Let X and Y be independent (strictly positive) gamma random variabl...
- 8.4.10: Let X and Y be independent (strictly positive) exponential random v...
Solutions for Chapter 8.4: Transformations of Two Random Variables
Full solutions for Fundamentals of Probability, with Stochastic Processes | 3rd Edition
See Arithmetic mean.
An estimator for a parameter obtained from a Bayesian method that uses a prior distribution for the parameter along with the conditional distribution of the data given the parameter to obtain the posterior distribution of the parameter. The estimator is obtained from the posterior distribution.
A chart used to organize the various potential causes of a problem. Also called a ishbone diagram.
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 probability of an event given that the random experiment produces an outcome in another event.
The probability 1?a associated with a conidence interval expressing the probability that the stated interval will contain the true parameter value.
An estimator that converges in probability to the true value of the estimated parameter as the sample size increases.
A tabular arrangement expressing the assignment of members of a data set according to two or more categories or classiication criteria
A correction factor used to improve the approximation to binomial probabilities from a normal distribution.
Continuous uniform random variable
A continuous random variable with range of a inite interval and a constant probability density function.
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.
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 .
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.
Cumulative sum control chart (CUSUM)
A control chart in which the point plotted at time t is the sum of the measured deviations from target for all statistics up to time t
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.
A probability distribution for a discrete random variable
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.
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 function used in the probability density function of a gamma random variable that can be considered to extend factorials