 8.8.1: Discrete randorr1 vectors X = [xi x;2 x3 J ' and Y = [ Y1 Y2 y3] ' ...
 8.8.2: Use t11e corr1ponents of Y = [Y 1 , ... , Y4]' in Exarr1ple 8.2 to ...
 8.8.3: (A) A test of ligr1t bulbs produced by a rnachine has three possibl...
 8.8.4: The tlireedirnensional raridom vector X = [X1 X2 X3]1 has PDF f x ...
 8.8.5: Z is the tV\rodirnensior1al standard norrnal rar1dom vector. The G...
 8.8.6: The daily noon ternperat11re, rneast1red in degrees Fahrenheit , in...
 8.8.7: For random variables X1, ... , X n in 5.10.3, let X = [ X1 X n] '. ...
 8.8.8: Random vector X has P DF fx (x ) = {ca'x 0 < x < 1, 0 other,vise, w...
 8.8.9: Given fx(x ) 'vith c = 2/ 3 and a1 = a2 = a3 = 1 in P roblem 8.1.2,...
 8.8.10: X = [X1 X2 X3]' has PDF . {6 fx (x) = 0 0 < X 1 < X2 < X3 < 1, othe...
 8.8.11: A 'vireless data terminal has three messages 'vaiting for transmiss...
 8.8.12: From the joint Pl\/IF PK(k) in 8.1.5, find the marginal P lVIFs (a)...
 8.8.13: Let N be the r dimensional random vector 'vith the multinomial PlV...
 8.8.14: T he random variables Y1, ... , Y4 have the joint PDF 0 <YI < Y2 < ...
 8.8.15: _As a generalization of the message transmission system in 8.1.5, c...
 8.8.16: The n, components Xi of random vector X have E[Xi] = 0 Var [Xi] = r...
 8.8.17: The 4dimensional random vector X has PDF fx(x ) = { ~ 0 <xi< l ,i ...
 8.8.18: As in Example 8.1, the random vector X has PDF fx(x) = { ~e'x x > ...
 8.8.19: The PDF of the 3dimensional random vector X is fx(x ) = { ~x, 0 <...
 8.8.20: The random vector X has PDF fx(x) = { ~xo 0 < X1 < X2 < X3, otherw...
 8.8.21: Discrete random vector X has PMF Px(x). Prove that for an invertibl...
 8.8.22: In the message transmission problem, 8.1.5, the PMF for the number ...
 8.8.23: In an automatic geolocation system, a dispatcher sends a message to...
 8.8.24: Let X1, . .. ,Xn denote n, iid random variables with P DF f x(x) an...
 8.8.25: Random variables X 1 and X 2 have zero expected value and variances...
 8.8.26: Let X 1, . .. , X n be iid random variables 'vith expected value 0,...
 8.8.27: The twodimensional random vector X and the threedimensional rando...
 8.8.28: T he fourdimensional random vector X has PDF fx(x) = { ~ 0<xi< 1,i...
 8.8.29: The random vector Y = [Y1 Y2 J' has covariance matrix C y = [ :; J]...
 8.8.30: In the message transmission system in Proble1n 8.1.5, the solution ...
 8.8.31: In the message transmission system in 8.1.5, p3 (1  p)k3 3; ki < ...
 8.8.32: Random vector X = [ X 1 PDF X1 > 0, ::e2 > 0, other\vise. (a) Find ...
 8.8.33: .As in Quiz 5.10 and Example 5.23, the 4dimensional random vector ...
 8.8.34: X = [X1 X2] ' is a random vector 'vith E[X] = [O O J' and covarianc...
 8.8.35: The t\vodimensional random vector Y has PDF f y (y ) = {~ y > o,[1...
 8.8.36: Let X be a random vector \vith correlation matrix R x and covarianc...
 8.8.37: X is the 3dimensional Gaussian random vector with expected value x...
 8.8.38: X = [X1 X2]' is the Gaussian random vector \vi th E[X] = [ 0 0 J' a...
 8.8.39: Given the Gaussian random vector X in 8.5.1, Y = AX + b , where 1/2...
 8.8.40: Let X be a Gaussian ( x , Cx) rando1n vector. Given a vector a , fi...
 8.8.41: Random variables X1 and X2 have zero expected value. The random vec...
 8.8.42: The Gaussian random vector X = [X 1 X2]' has expected value E [X] =...
 8.8.43: The Gaussian random vector X = [X1 X2]' has expected value E [X] = ...
 8.8.44: Let X be a Gaussian random vector v;,rith expected value [/Li /.lz ...
 8.8.45: X = [X1 X2] ' is a Gaussian random vector 'vi th E[X] = [ 0 0 J' an...
 8.8.46: Let X be a Gaussian ( x , C x ) random vector. Let Y = AX where A i...
 8.8.47: T he 2 x 2 inatrix [ cos e Q = sine  sin el cos e is called a rota...
 8.8.48: X = [X1 X2]' is a Gaussian (0, C x ) vector where Thus, depending o...
 8.8.49: An 77,dimensional Gaussian vector W has a block diagonal covarianc...
 8.8.50: In this problem, we extend the proof of Theorem 8.11 to the case wh...
 8.8.51: Consider the vector X in 8.5.l and define Y = (X1 +X2 +X3)/3. \i\fh...
 8.8.52: A better model for the sailboat race of I>roblem 5.10.8 accounts fo...
 8.8.53: For the vector of daily temperatures [T1 T31J 1 and average tempera...
 8.8.54: vVe continue 8.6.2 'vhere the vector X of finish times has correlat...
 8.8.55: Write a l\IIATLAB program that simulates m, runs of the weekly lott...
Solutions for Chapter 8: Random Vectors
Full solutions for Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers  3rd Edition
ISBN: 9781118324561
Solutions for Chapter 8: Random Vectors
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers, edition: 3. Chapter 8: Random Vectors includes 55 full stepbystep solutions. Since 55 problems in chapter 8: Random Vectors have been answered, more than 11091 students have viewed full stepbystep solutions from this chapter. Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers was written by and is associated to the ISBN: 9781118324561.

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

Acceptance region
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

Addition rule
A formula used to determine the probability of the union of two (or more) events from the probabilities of the events and their intersection(s).

Alias
In a fractional factorial experiment when certain factor effects cannot be estimated uniquely, they are said to be aliased.

Analysis of variance (ANOVA)
A method of decomposing the total variability in a set of observations, as measured by the sum of the squares of these observations from their average, into component sums of squares that are associated with speciic deined sources of variation

Attribute
A qualitative characteristic of an item or unit, usually arising in quality control. For example, classifying production units as defective or nondefective results in attributes data.

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).

Components of variance
The individual components of the total variance that are attributable to speciic sources. This usually refers to the individual variance components arising from a random or mixed model analysis of variance.

Convolution
A method to derive the probability density function of the sum of two independent random variables from an integral (or sum) of probability density (or mass) functions.

Covariance
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 .

Deming
W. Edwards Deming (1900–1993) was a leader in the use of statistical quality control.

Dispersion
The amount of variability exhibited by data

Error sum of squares
In analysis of variance, this is the portion of total variability that is due to the random component in the data. It is usually based on replication of observations at certain treatment combinations in the experiment. It is sometimes called the residual sum of squares, although this is really a better term to use only when the sum of squares is based on the remnants of a modelitting process and not on replication.

Error variance
The variance of an error term or component in a model.

Estimate (or point estimate)
The numerical value of a point estimator.

Experiment
A series of tests in which changes are made to the system under study

Exponential random variable
A series of tests in which changes are made to the system under study

Factorial experiment
A type of experimental design in which every level of one factor is tested in combination with every level of another factor. In general, in a factorial experiment, all possible combinations of factor levels are tested.

Forward selection
A method of variable selection in regression, where variables are inserted one at a time into the model until no other variables that contribute signiicantly to the model can be found.

Gaussian distribution
Another name for the normal distribution, based on the strong connection of Karl F. Gauss to the normal distribution; often used in physics and electrical engineering applications