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### assignment8

Course: EE 101, Spring 2012
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505 EE B Fall 2011 Assignment 8 For full credit, you must show all of the steps or reasoning that you used to get the answer. 1. A joint probability density function is given as c e x1 x1 &gt; 0 and | x2 | &lt; x1 0 f X1 X2 ( x 1 , x 2 ) = otherwise (a) Draw the region for which the joint probability density function is non-zero. (b) Determine the value of c. (c) Find and sketch the marginal...

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505 EE B Fall 2011 Assignment 8 For full credit, you must show all of the steps or reasoning that you used to get the answer. 1. A joint probability density function is given as c e x1 x1 > 0 and | x2 | < x1 0 f X1 X2 ( x 1 , x 2 ) = otherwise (a) Draw the region for which the joint probability density function is non-zero. (b) Determine the value of c. (c) Find and sketch the marginal probability density functions. (d) Find the joint cumulative distribution function. (e) Evaluate Pr(X1 1). (f) Evaluate Pr(X1 + X2 < 1). (g) Evaluate E{ X1 X2 }. (h) Are X1 and X2 independent? Make sure that you justify answer. (i) your Find and sketch f X1 | X2 ( x1 ) and f X2 | X1 ( x2 ). (j) Find and sketch f X1 |{ X2 >1} ( x1 ). (k) Let Y1 = X1 + X2 Y2 = X1 X2 Find an expression for the joint probability density function for Y1 and Y2 . Are Y1 and Y2 independent? 2. The probability that the rst packet arrives at a certain network node at a time X given that the node comes online at a time Y is f X |Y ( x ) = e ( x y) 0yx< 0 x<y The probability that the node comes online at time Y is uniformly distributed between 0 and 1. (a) Find the joint probability density function f XY ( x, y). (b) Find f X ( x ). 1
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Alaska Anch - EE - 101
EE 505 B Fall 2011Assignment 8 SolutionsFor full credit, you must show all of the steps or reasoning that you used to get theanswer.1. (a) The denition of the region can be written asB = cfw_( x1 , x2 ) : x1 &gt; 0, x1 &lt; x2 &lt; x1 .The region is shown i
Alaska Anch - EE - 101
1.4 The Matrix Equation Ax bLinear combinations can be viewed as a matrix-vectormultiplication.DefinitionIf A is an m n matrix, with columns a 1 , a 2 , , a n , and if x is inR n , then the product of A and x, denoted by Ax, is the linearcombination
Alaska Anch - EE - 101
Delta Function - from Wolfram MathWorld1 of 4http:/mathworld.wolfram.com/DeltaFunction.htmlSEARCH MATHWORLDAlgebraApplied MathematicsCalculus and AnalysisDiscrete MathematicsFoundations of MathematicsGeometryDelta Function in theCalculus and An
Alaska Anch - EE - 101
DFT DefinitionPage 1 of 2Next | Prev | Up | Top | Index | JOS Index | JOS Pubs | JOS Home | SearchDFT DefinitionThe Discrete Fourier Transform (DFT) of a signalwhere `may be defined by' means `is defined as' or `equals by definition', andThe sampl
Alaska Anch - EE - 101
Divergence Theorem ExamplesGauss' divergence theorem relates triple integrals and surface integrals.GAUSS' DIVERGENCE THEOREM Let F be a vector field. Let W be a closed surface, and let e be the region inside of W . Then: ( ( F . A oe ( ( ( divaFb .ZW
Alaska Anch - EE - 101
Section 17.9 The Divergence TheoremTurning a Flux Integral into a Triple Integral The last result we consider is a generalization of Greens Theorem converting a ux integral over a closed surface into a triple integral over the interior of the surface. 1.
Alaska Anch - EE - 101
Alaska Anch - EE - 101
EIGENVALUES AND EIGENVECTORS1. DefinitionThey are dened in terms of each other. Let A be an n n matrix. A vectorv = 0 is an eigenvector of A with eigenvalue if the equationAv = vis satised.Note that eigenvectors are not uniquely dened:If v is an ei
Alaska Anch - EE - 101
Euler's IdentityPage 1 of 1Next | Prev | Up | Top | Index | JOS Index | JOS Pubs | JOS Home | SearchEuler's IdentityEuler's identity (or `theorem' or `formula') is(Euler's Identity)To `prove' this, we will first define what we mean by `, is assumed
Alaska Anch - EE - 101
453.701 Linear Systems, S.M. Tan, The University of Auckland9-1Chapter 9 The Discrete Fourier transform9.1DenitionWhen computing spectra on a computer it is not possible to carry out the integrals involved inthe continuous time Fourier transform. In
Alaska Anch - EE - 101
EE 505 B Fall 2011Final ExamNota bene: For full credit you must show all of the steps or reasoning that you used toget the answer.1. [35 points] U is a uniformly distributed random variable with probability densityfunction1 0u1f U (u) =0 for all o
Alaska Anch - EE - 101
EE 505 B Fall 2011Final Exam Solutions1. [35 points](a) [10 points] We use the formulaf X (x) =f U (u)ddu g(u) u = u iixThere is only one solution u1 that sat1ises the equation x = ln(u):xu1 = e xexSubstituting u1 into the pdf:f U ( u1 ) =
Alaska Anch - EE - 101
Fourier Transform - from Wolfram MathWorld1 of 5http:/mathworld.wolfram.com/FourierTransform.htmlSEARCH MATHWORLDAlgebraApplied MathematicsCalculus and AnalysisDiscrete MathematicsFoundations of MathematicsGeometryFourier Transform in theCalcul
Alaska Anch - EE - 101
The Eigenvalue Problems - 8.81. Definition of Eigenvalues and Eigenvectors:Let A be an n ! n matrix. A scalar ! is said to be an eigenvalue of A if the linear system Av &quot; !v has anonzero solution vector v. The solution vector v is said to be an eigenve
Alaska Anch - EE - 101
List of integrals of trigonometric functions - Wikipedia, the free encyclopediaPage 1 of 10List of integrals of trigonometric functionsFrom Wikipedia, the free encyclopediaThe following is a list of integrals (antiderivativefunctions) of trigonometri
Alaska Anch - EE - 101
List of trigonometric identities - Wikipedia, the free encyclopediaPage 1 of 25List of trigonometric identitiesFrom Wikipedia, the free encyclopediaIn mathematics, trigonometric identities areequalities that involve trigonometric functions andare tr
Alaska Anch - EE - 101
Mathematics of the DFTPage 1 of 3Next | Prev | Up | Top | Index | JOS Index | JOS Pubs | JOS Home | SearchMathematics of the DFTIn the signal processing literature, it is common to write the DFT and its inverse in the more pure formbelow, obtained by
Alaska Anch - EE - 101
Edward Neuman Department of Mathematics Southern Illinois University at Carbondale edneuman@siu.eduOne of the nice features of MATLAB is its ease of computations with vectors and matrices. In this tutorial the following topics are discussed: vectors a
Alaska Anch - EE - 101
EE 505 B Fall 2011Midterm ExamNota bene: For full credit you must show all of the steps or reasoning that you used toget the answer.1. [23 points](a) [3 points] If one appends a column vector b to a matrix A, then the column.space of the matrix get
Alaska Anch - EE - 101
EE 505 B Fall 2011Midterm Exam Solutions1. [23 points](a) [3 points] b is in the column space of the matrix, or (equivalently) b is alinear combination of the columns of A(b) (i) [6 points] The eigenvalues are found using the characteristic equation:
Alaska Anch - EE - 101
Chapter 04.02VectorsAfter reading this chapter, you should be able to:1.2.3.4.5.define a vector,add and subtract vectors,find linear combinations of vectors and their relationship to a set of equations,explain what it means to have a linearly i
Alaska Anch - EE - 101
LECTURE 2Orthogonal Vectors andMatricesOBJECTIVE: Orthogonality is central to many ofthe main algorithms on linear algebra. We review theingredients: orthogonal vectors and matrices.2-1Transp oseDenitionThe transpose AT of an m n matrix A is n m
Alaska Anch - EE - 101
Orthogonality TutorialPage 1 of 7ORTHOGONALITY TUTORIALTUTORIALSTUTORIALS HOMEGENERAL MATHNOTATION &amp;METHODS OFPROOFINDUCTIONCOMPLEXNUMBERSPOLYNOMIALSOrthogonal SetsVectors v, u are orthogonal or perpendicular to each other if vvLINEAR ALGE
Alaska Anch - EE - 101
Starting with Two MatricesGilbert Strang, Massachusetts Institute of TechnologyImagine that you have never seen matrices. On the principle that examples are amazingly powerful, westudy two matrices A and C . The reader is requested to be exceptionally
Alaska Anch - EE - 101
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UCLA - MATH - 172
UCLA - MATH - 172
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UCLA - MATH - 172
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UCLA - MATH - 172
UCLA - MATH - 172
Using the TI BA II Plus for Actuarial Finance CalculationsIntroduction. This manual is being written to help actuarial students becomemore efficient problem solvers for the Part II examination of the CasualtyActuarial Society and the Society of Actuari
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FINS3635 S2/2011Put-Call ParityMatthias ThulLast Update: September 14, 2011This documents shows you how a the general put-call relationship for European optionscan be obtained by simple no-arbitrage arguments and gives some examples of how it can be
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