An Introduction To Information Theory – Fazlollah M Reza

1 / 2 ax exp – 9(‘ C — a)2 + exp — ^ ( — X — a)2 p(y) = 2 \/2-iray exp y — — a a + exp y — — a a y > 0 Multidimensional Case. Let (Xi,X2, . . . ,Xn) be an n-dimensional random variable with CDF F(xi,x2, . . .

,xn). Let yi — ffi(Xi,x2, . . . ,x„) F2 = g2(Xi,X2, . . . ,X„) Yn = ?n(XX,X2, . . . ,Xn) (5-74) It can be shown without difficulty that a generalization of the rela¬ tion (5-73) is valid under certain appropriate circumstances. First of all, the transformation of Eqs. (5-74) must be one-to-one, and all the gic must be differentiable and have continuous Furthermore, dY, 3Fj dYi dX1 ax2 ax„ <9F2 ar2 au2 dXi ax2 ax„ 6Yn dYn 9Yn dXx ax2 dXn 5* 0 (5-75) In the second place, the variable (Xi,X2, .

. . ,Xn) must possess a density function/(xi,x2, . . . ,x„). Under such assumptions, the density function of (Fi,F2, . . . , F„) can be derived as a logical extension of Eqs. (5-73). pQ/i,i/2, ■ • – ,Vn) = j^f(xi,x2, . . . ,xn) (5-76) Example 6-11. Let the density function for the random variable (X\,X2) be Find the density function for the variable (Fi,F2), where Solution.

According to Eq. (5-75), Also note that Hence. g(y 1,2/2) The simplicity of this problem is due to the linear transformation of the variable. Otherwise, A would not remain invariant for all values of the variable. Multivalued Transformations. Let F = g(X) be any real continuous transformation defined for all possible values of the random variable X. The probability distribution for the random variable F is G(y) = P{g(X) < y} (5-77) For convenience of analysis, the range of the values of x may be divided into pieces in which y = g{x) is monotonic either nonincreasing or FlG- 5'9- ^ multivalued transformation , .

TTT.,. , . of the random variable, nondecreasing. Without loss of generality, we assume that the functional relationship y — g{x) is such that in each of the above intervals y is differentiable, single-valued, and has a nonzero derivative.

Ahrendt and Savant • Servomechanism Practice Angelo • Electronic Circuits Aseltine • Transform Method in Linear System Analysis Atwater • Introduction to Microwave Theory Bailey and Gault • Alternating-current Machinery Beranek • Acoustics Bracewell ■ The Fourier Transform and Its Application Brenner and Javid • Analysis of Electric Circuits Brown • Analysis of Linear Time-invariant Systems Bruns and Saunders • Analysis of Feedback Control Systems Cage • Theory and Application of Industrial Electronics Cauer • Synthesis of Linear Communication Networks Chen ■ The Analysis of Linear Systems Chen • Linear Network Design and Synthesis Chirlian • Analysis and Design of Electronic Circuits Chirlian and Zemanian • Electronics Clement and Johnson • Electrical Engineering Science Cote and Oakes • Linear Vacuum-tube and Transistor Circuits Cuccia • Harmonics, Sidebands, and Transients in Communication Engineering Cunningham • Introduction to Nonlinear Analysis D’Azzo and Houpis • Feedback Control System Analysis and Synthesis Eastman • Fundamentals of Vacuum Tubes Elgerd • Control Systems Theory Feinstein • Foundations of Information Theory Fitzgerald, Higginbotham, and Giiabel • Basic Electrical Engineering Fitzgerald and Kingsley • Electric Machinery Frank • Electrical Measurement Analysis Friedland, Wing, and Ash • Principles of Linear Networks Gehmlich and Hammond • Electromechanical Systems Ghausi • Principles and Design of Linear Active Circuits Ghose • Microwave Circuit Theory and Analysis Greiner • Semiconductor Devices and Applications Hammond ■ Electrical Engineering Hancock ■ An Introduction to the Principles of Communication Theory Happell and Hesselberth ■ Engineering Electronics Harman Fundamentals of Electronic Motion Harman • Principles of the Statistical Theory of Communication Harman and Lytle ■ Electrical and Mechanical Networks Harrington Introduction to Electromagnetic Engineering Harrington • Time-harmonic Electromagnetic Fields Hayashi • Nonlinear Oscillations in Physical Systems Hayt • Engineering Electromagnetics Hayt and Kemmerly • Engineering Circuit Analysis Hill • Electronics in Engineering Javid and Brenner • Analysis, Transmission, and Filtering of Signals Javid and Brown • Field Analysis and Electromagnetics Johnson • Transmission Lines and Networks Koenig and Blackwell • Electromechanical System Theory Koenig, Tokad, and Kesavan Analysis of Discrete Physical Systems Kraus • Antennas Kraus • Electromagnetics Kuh and Pederson • Principles of Circuit Synthesis Kuo • Linear Networks and Systems Ledley ■ Digital Computer and Control Engineering LePage • Analysis of Alternating-current Circuits LkPage • Complex Variables and the Laplace Transform for Engineering LePage and Seely • General Network Analysis Levi and Panzer • Electromechanical Power Conversion Ley, Lutz, and Rehberg ■ Linear Circuit Analysis Linvill and Gibbons • Transistors and Active Circuits Littauer ■ Pulse Electronics Lynch and Truxal • Introductory System Analysis Lynch and Truxal • Principles of Electronic Instrumentation Lynch and Truxal • Signals and Systems in Electrical Engineering Manning • Electrical Circuits McCluskey • Introduction to the Theory of Switching Circuits Meisel • Principles of Electromechanical-energy Conversion Millman • Vacuum-tube and Semiconductor Electronics Millman and Seely • Electronics Millman and Taub ■ Pulse and Digital Circuits Millman and Taub Pulse, Digital, and Switching Waveforms Mishkin and Braun • Adaptive Control Systems Moore • Traveling-wave Engineering Nanavati An Introduction to Semiconductor Electronics Pettit • Electronic Switching, Timing, and Pulse Circuits Pettit and McWhorter • Electronic Amplifier Circuits Pfeiffer • Concepts of Probability Theory Pfeiffer • Linear Systems Analysis IIeza • An Introduction to Information Theory Reza and Seely • Modern Network Analysis Rogers Introduction to Electric Fields Ruston and Bordogna ■ Electric Networks: Functions, Filters, Analysis Ryder • Engineering Electronics Schwartz • Information Transmission, Modulation, and Noise Schwarz and Friedland • Linear Systems Seely • Electromechanical Energy Conversion Seely • Electron-tube Circuits Seely • Electronic Engineering Seely ■ Introduction to Electromagnetic Fields Seely • Radi

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