 11.3.1: According to the Bureau of Engraving and Printing, http://www.money...
 11.3.2: Show that if for a nonnegative random variable X, P (X < 2) = 3/5, ...
 11.3.3: Let X be a nonnegative random variable with E(X) = 5 and E(X2) = 42...
 11.3.4: The average and standard deviation of lifetimes of light bulbs manu...
 11.3.5: Suppose that the average number of accidents at an intersection is ...
 11.3.6: The average IQ score on a certain campus is 110. If the variance of...
 11.3.7: The waiting period from the time a book is ordered until it is rece...
 11.3.8: Show that for a nonnegative random variable X with mean , P (X 2) 1/2.
 11.3.9: Suppose that X is a random variable with E(X) = Var(X) = . What doe...
 11.3.10: From a distribution with mean 42 and variance 60, a random sample o...
 11.3.11: The mean IQ of a randomly selected student from a specific universi...
 11.3.12: For a distribution, the mean of a random sample is taken as estimat...
 11.3.13: To determine p, the proportion of time that an airline operator is ...
 11.3.14: For a coin, p, the probability of heads is unknown. To estimate p, ...
 11.3.15: Let X be a random variable with mean . Show that if E 4 (X )2n 5 < ...
 11.3.16: Let X be a random variable and k be a constant. Prove that P (X > t...
 11.3.17: Prove that if the random variables X and Y satisfy E 4 (X Y )2 5 = ...
 11.3.18: Let X and Y be two randomly selected numbers from the set of positi...
 11.3.19: Let X be a random variable; show that for > 1 and t > 0, P * X 1 t ...
 11.3.20: Let the probability density function of a random variable X be f (x...
 11.3.21: Let {x1, x2, . . . , xn} be a set of real numbers and define x = 1 ...
Solutions for Chapter 11.3: Markov and Chebyshev Inequalities
Full solutions for Fundamentals of Probability, with Stochastic Processes  3rd Edition
ISBN: 9780131453401
Solutions for Chapter 11.3: Markov and Chebyshev Inequalities
Get Full SolutionsFundamentals of Probability, with Stochastic Processes was written by and is associated to the ISBN: 9780131453401. Chapter 11.3: Markov and Chebyshev Inequalities includes 21 full stepbystep solutions. Since 21 problems in chapter 11.3: Markov and Chebyshev Inequalities have been answered, more than 14106 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Fundamentals of Probability, with Stochastic Processes, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions.

2 k p  factorial experiment
A fractional factorial experiment with k factors tested in a 2 ? p fraction with all factors tested at only two levels (settings) each

`error (or `risk)
In hypothesis testing, an error incurred by rejecting a null hypothesis when it is actually true (also called a type I error).

Average
See Arithmetic mean.

Chance cause
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.

Coeficient of determination
See R 2 .

Conditional probability distribution
The distribution of a random variable given that the random experiment produces an outcome in an event. The given event might specify values for one or more other random variables

Conidence interval
If it is possible to write a probability statement of the form PL U ( ) ? ? ? ? = ?1 where L and U are functions of only the sample data and ? is a parameter, then the interval between L and U is called a conidence interval (or a 100 1( )% ? ? conidence interval). The interpretation is that a statement that the parameter ? lies in this interval will be true 100 1( )% ? ? of the times that such a statement is made

Contrast
A linear function of treatment means with coeficients that total zero. A contrast is a summary of treatment means that is of interest in an experiment.

Correlation
In the most general usage, a measure of the interdependence among data. The concept may include more than two variables. The term is most commonly used in a narrow sense to express the relationship between quantitative variables or ranks.

Correlation coeficient
A dimensionless measure of the linear association between two variables, usually lying in the interval from ?1 to +1, with zero indicating the absence of correlation (but not necessarily the independence of the two variables).

Critical value(s)
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

Defect concentration diagram
A quality tool that graphically shows the location of defects on a part or in a process.

Defectsperunit control chart
See U chart

Dependent variable
The response variable in regression or a designed experiment.

Design matrix
A matrix that provides the tests that are to be conducted in an experiment.

Discrete distribution
A probability distribution for a discrete random variable

Empirical model
A model to relate a response to one or more regressors or factors that is developed from data obtained from the system.

Erlang random variable
A continuous random variable that is the sum of a ixed number of independent, exponential random variables.

Fixed factor (or fixed effect).
In analysis of variance, a factor or effect is considered ixed if all the levels of interest for that factor are included in the experiment. Conclusions are then valid about this set of levels only, although when the factor is quantitative, it is customary to it a model to the data for interpolating between these levels.