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1 | (6) |
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What Is Chance, and Why Study It? |
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3 | (4) |
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3 | (1) |
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Probability Problems in Optics |
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4 | (1) |
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Statistical Problems in Optics |
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5 | (1) |
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5 | (2) |
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7 | (32) |
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Notion of an Experiment; Events |
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7 | (2) |
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Event Space; The Space Event |
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8 | (1) |
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8 | (1) |
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9 | (1) |
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Definition of Probability |
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9 | (1) |
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Relation to Frequency of Occurrence |
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10 | (1) |
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Some Elementary Consequences |
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10 | (2) |
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11 | (1) |
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11 | (1) |
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12 | (1) |
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The ``Traditional'' Definition of Probability |
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12 | (1) |
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Illustrative Problem: A Dice Game |
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13 | (1) |
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Illustrative Problem: Let's (Try to) Take a Trip |
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14 | (1) |
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15 | (1) |
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Optical Objects and Images as Probability Laws |
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15 | (2) |
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17 | (1) |
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The Quantity of Information |
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18 | (2) |
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20 | (2) |
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Illustrative Problem: Let's (Try to) Take a Trip (Continued) |
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21 | (1) |
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22 | (1) |
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22 | (1) |
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``Additivity'' Property of Information |
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23 | (1) |
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23 | (1) |
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Illustrative Problem: Transmittance Through a Film |
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24 | (1) |
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How to Correct a Success Rate for Guesses |
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25 | (1) |
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26 | (1) |
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Some Optical Applications |
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27 | (1) |
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Information Theory Application |
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28 | (1) |
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Application to Markov Events |
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28 | (1) |
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29 | (1) |
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What is the Probability of Winning a Lottery Jackpot? |
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30 | (1) |
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What is the Probability of a Coincidence of Birthdays at a Party? |
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31 | (8) |
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Continuous Random Variables |
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39 | (40) |
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Definition of a Random Variable |
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39 | (1) |
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Probability Density Function, Basic Properties |
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39 | (2) |
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Information Theory Application: Continuous Limit |
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41 | (1) |
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Optical Application: Continuous Form of Imaging Law |
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41 | (1) |
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42 | (1) |
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Optical Application: Moments of the Slit Diffraction Pattern |
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43 | (1) |
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Information Theory Application |
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44 | (1) |
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Case of Statistical Independence |
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45 | (1) |
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45 | (1) |
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46 | (1) |
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Deterministic Limit; Representations of the Dirac δ-Function |
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47 | (1) |
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Correspondence Between Discrete and Continuous Cases |
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48 | (1) |
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48 | (1) |
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The Means of an Algebraic Expression: A Simplified Approach |
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49 | (1) |
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A Potpourri of Probability Laws |
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50 | (18) |
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50 | (1) |
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51 | (1) |
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51 | (1) |
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52 | (1) |
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53 | (1) |
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53 | (2) |
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Normal (Multi-Dimensional) |
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55 | (1) |
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Skewed Gaussian Case; Gram-Charlier Expansion |
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56 | (1) |
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56 | (2) |
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58 | (1) |
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58 | (1) |
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58 | (10) |
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Derivation of Heisenberg Uncertainty Principle |
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68 | (2) |
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Schwarz Inequality for Complex Functions |
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68 | (1) |
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68 | (1) |
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69 | (1) |
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Hirschman's Form of the Uncertainty Principle |
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70 | (1) |
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70 | (3) |
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Kullback-Leibler Information |
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70 | (1) |
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71 | (1) |
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71 | (1) |
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72 | (1) |
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72 | (1) |
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72 | (1) |
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Fisher Information Matrix |
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73 | (6) |
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Fourier Methods in Probability |
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79 | (28) |
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Characteristic Function Defined |
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79 | (1) |
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Use in Generating Moments |
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80 | (1) |
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An Alternative to Describing RV x |
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80 | (1) |
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80 | (1) |
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81 | (1) |
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81 | (1) |
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82 | (1) |
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82 | (1) |
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82 | (1) |
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Normal Case (One Dimension) |
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83 | (1) |
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83 | (1) |
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Normal Case (Two Dimensions) |
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83 | (1) |
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Convolution Theorem, Transfer Theorem |
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83 | (1) |
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Probability Law for the Sum of Two Independent RV's |
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84 | (1) |
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85 | (2) |
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Imaging Equation as the Sum of Two Random Displacements |
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85 | (1) |
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85 | (2) |
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Sum of n Independent RV's; The ``Random Walk'' Phenomenon |
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87 | (2) |
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Resulting Mean and Variance: Normal, Poisson, and General Cases |
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89 | (1) |
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89 | (1) |
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Case of Two Gaussian Bivariate RV's |
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90 | (1) |
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Sampling Theorems for Probability |
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91 | (1) |
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Case of Limited Range of x, Derivation |
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91 | (1) |
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92 | (1) |
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93 | (1) |
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Case of Limited Range of ω |
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94 | (1) |
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94 | (1) |
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95 | (2) |
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How Large Does n Have To Be? |
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97 | (1) |
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97 | (3) |
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Cascaded Electro-Optical Systems |
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97 | (1) |
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98 | (1) |
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99 | (1) |
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Generating Normally Distributed Numbers from Uniformly Random Numbers |
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100 | (2) |
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102 | (5) |
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Functions of Random Variables |
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107 | (40) |
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Case of a Single Random Variable |
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107 | (1) |
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108 | (1) |
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Application from Geometrical Optics |
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109 | (1) |
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110 | (1) |
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111 | (1) |
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Case of n Random Variables, r Roots |
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111 | (1) |
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112 | (1) |
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112 | (1) |
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Application of Transformation Theory to Laser Speckle |
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113 | (6) |
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113 | (1) |
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114 | (1) |
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114 | (1) |
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Marginal Probabilities for Light Amplitudes Ure, Uim |
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115 | (1) |
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Correlation Between Ure and Uim |
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116 | (1) |
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Joint Probability Law for Ure, Uim |
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117 | (1) |
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Probability Laws for Intensity and Phase; Transformation of the RV's |
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117 | (1) |
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Marginal Laws for Intensity and Phase |
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118 | (1) |
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Signal-to-Noise (S/N) Ratio in the Speckle Image |
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118 | (1) |
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Speckle Reduction by Use of a Scanning Aperture |
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119 | (4) |
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119 | (1) |
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Probability Density for Output Intensity p1(v) |
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120 | (1) |
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121 | (1) |
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Standard Form for the Chi-Square Distribution |
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122 | (1) |
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Calculation of Spot Intensity Profiles Using Transformation Theory |
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123 | (3) |
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124 | (1) |
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Implementation by Ray-Trace |
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125 | (1) |
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Application of Transformation Theory to a Satellite-Ground Communication Problem |
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126 | (14) |
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Unequal Numbers of Input and Output Variables: ``Helper Variables'' |
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140 | (2) |
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Probability Law for a Quotient of Random Variables |
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140 | (1) |
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Probability Law for a Product of Independent Random Variables |
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141 | (1) |
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More Complicated Transformation Problems |
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142 | (1) |
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Use of an Invariance Principle to Find a Probability Law |
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142 | (2) |
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Probability Law for Transformation of a Discrete Random Variable |
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144 | (3) |
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Bernoulli Trials and Limiting Cases |
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147 | (28) |
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147 | (2) |
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149 | (3) |
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Illustrative Problem: Let's (Try to) Take a Trip: The Last Word |
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149 | (1) |
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Illustrative Problem: Mental Telepathy as a Communication Link? |
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150 | (2) |
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Characteristic Function and Moments |
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152 | (1) |
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Optical Application: Checkerboard Model of Granularity |
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152 | (2) |
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154 | (3) |
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154 | (1) |
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Example of Degree of Approximation |
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155 | (1) |
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Normal Limit of Poisson Law |
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156 | (1) |
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Optical Application: The Shot Effect |
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157 | (1) |
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Optical Application: Combined Sources |
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158 | (1) |
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Poisson Joint Count for Two Detectors-Intensity Interferometry |
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158 | (4) |
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The Normal Limit (De Moivre-Laplace Law) |
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162 | (13) |
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162 | (1) |
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163 | (1) |
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Use of the Error Function |
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164 | (11) |
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The Monte Carlo Calculation |
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175 | (16) |
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Producing Random Numbers That Obey a Prescribed Probability Law |
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176 | (2) |
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177 | (1) |
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177 | (1) |
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Analysis of the Photographic Emulsion by Monte Carlo Calculation |
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178 | (2) |
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Application of the Monte Carlo Calculation to Remote Sensing |
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180 | (1) |
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Monte Carlo Formation of Optical Images |
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181 | (2) |
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182 | (1) |
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Monte Carlo Simulation of Speckle Patterns |
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183 | (8) |
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191 | (52) |
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Definition of a Stochastic Process |
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191 | (1) |
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Definition of Power Spectrum |
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192 | (2) |
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Some Examples of Power Spectra |
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194 | (1) |
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Definition of Autocorrelation Function; Kinds of Stationarity |
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194 | (1) |
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Fourier Transform Theorem |
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195 | (1) |
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Case of a ``White'' Power Spectrum |
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196 | (1) |
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Application: Average Transfer Function Through Atmospheric Turbulence |
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197 | (4) |
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Statistical Model for Phase Fluctuations |
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198 | (1) |
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A Transfer Function for Turbulence |
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199 | (2) |
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Transfer Theorems for Power Spectra |
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201 | (7) |
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Determining the MTF Using Random Objects |
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201 | (1) |
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Speckle Interferometry of Labeyrie |
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202 | (1) |
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Resolution Limits of Speckle Interferometry |
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203 | (5) |
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Transfer Theorem for Autocorrelation: The Knox-Thompson Method |
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208 | (3) |
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211 | (1) |
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212 | (1) |
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213 | (4) |
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217 | (4) |
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Definition of Restoring Filter |
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217 | (1) |
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218 | (1) |
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219 | (2) |
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Information Content in the Optical Image |
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221 | (5) |
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222 | (1) |
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223 | (1) |
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223 | (1) |
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224 | (1) |
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225 | (1) |
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225 | (1) |
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Data Information and Its Ability to be Restored |
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226 | (1) |
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Superposition Processes; the Shot Noise Process |
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227 | (16) |
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229 | (1) |
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229 | (1) |
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230 | (1) |
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231 | (1) |
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231 | (1) |
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232 | (1) |
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232 | (1) |
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Signal-to-Noise (S/N) Ratio |
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233 | (1) |
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234 | (1) |
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235 | (1) |
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Application: An Overlapping Circular Grain Model for the Emulsion |
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236 | (1) |
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Application: Light Fluctuations due to Randomly Tilted Waves, the ``Swimming Pool'' Effect |
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237 | (6) |
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Introduction to Statistical Methods: Estimating the Mean, Median, Variance, S/N, and Simple Probability |
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243 | (34) |
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Estimating a Mean from a Finite Sample |
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244 | (1) |
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244 | (1) |
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245 | (1) |
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246 | (1) |
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Error in a Discrete, Linear Processor: Why Linear Methods Often Fail |
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246 | (2) |
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Estimating a Probability: Derivation of the Law of Large Numbers |
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248 | (1) |
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249 | (1) |
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Illustrative Uses of the Error Expression |
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250 | (2) |
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Estimating Probabilities from Empirical Rates |
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250 | (1) |
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Aperture Size for Required Accuracy in Transmittance Readings |
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251 | (1) |
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Probability Law for the Estimated Probability; Confidence Limits |
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252 | (1) |
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Calculation of the Sample Variance |
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253 | (5) |
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Unbiased Estimate of the Variance |
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253 | (2) |
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Expected Error in the Sample Variance |
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255 | (1) |
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256 | (2) |
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Estimating the Signal-to-Noise Ratio; Student's Probability Law |
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258 | (3) |
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258 | (1) |
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259 | (2) |
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Limit c → 0; A Student Probability Law |
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261 | (1) |
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Properties of a Median Window |
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261 | (2) |
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263 | (6) |
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Probability Law for the Median |
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264 | (1) |
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Laser Speckle Case: Exponential Probability Law |
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264 | (5) |
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Dominance of the Cauchy Law in Diffraction |
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269 | (8) |
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Estimating an Optical Slit Position: An Optical Central Limit Theorem |
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270 | (1) |
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Analysis by Characteristic Function |
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270 | (2) |
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Cauchy Limit, Showing Independence to Aberrations |
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272 | (1) |
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Widening the Scope of the Optical Central Limit Theorem |
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273 | (4) |
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Introduction to Estimating Probability Laws |
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277 | (30) |
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Estimating Probability Densities Using Orthogonal Expansions |
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278 | (3) |
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281 | (1) |
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The Multinomial Probability Law |
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282 | (1) |
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282 | (1) |
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283 | (1) |
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Estimating an Empirical Occurrence Law as the Maximum Probable Answer |
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283 | (24) |
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Principle of Maximum Probability (MP) |
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284 | (1) |
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285 | (1) |
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The Search for ``Maximum Prior Ignorance'' |
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286 | (1) |
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Other Types of Estimates (Summary) |
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287 | (1) |
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Return to Maximum Entropy Estimation, Discrete Case |
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288 | (1) |
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Transition to a Continuous Random Variable |
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289 | (1) |
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290 | (1) |
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290 | (1) |
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Illustrative Example: Significance of the Normal Law |
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290 | (1) |
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The Smoothness Property; Least Biased Aspect |
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291 | (1) |
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A Well Known Distribution Derived |
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292 | (1) |
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When Does the Maximum Entropy Estimate Equal the True Law? |
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293 | (1) |
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Maximum Probable Estimates of Optical Objects |
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294 | (2) |
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Case of Nearly Featureless Objects |
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296 | (11) |
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The Chi-Square Test of Significance |
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307 | (14) |
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308 | (1) |
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Probability Law for χ2 Statistic |
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309 | (2) |
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311 | (1) |
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Equivalence of Chi-Square to Other Statistics; Sufficient Statistics |
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312 | (1) |
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313 | (1) |
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Generalization to N Voters |
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314 | (1) |
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315 | (6) |
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The Student t-Test on the Mean |
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321 | (12) |
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Cases Where Data Accuracy is Unknown |
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322 | (1) |
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Philosophy of the Approach: Statistical Inference |
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322 | (1) |
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323 | (2) |
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Student's t-Distribution: Derivation |
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325 | (1) |
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Some Properties of Student's t-Distribution |
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326 | (1) |
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Application to the Problem: Student's t-Test |
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327 | (1) |
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327 | (2) |
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329 | (4) |
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333 | (8) |
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Snedecor's F-Distribution; Derivation |
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333 | (1) |
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Some Properties of Snedecor's F-Distribution |
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334 | (1) |
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335 | (1) |
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336 | (1) |
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Application to Image Detection |
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336 | (5) |
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Least-Squares Curve Fitting-Regression Analysis |
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341 | (22) |
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Summation Model for the Physical Effect |
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341 | (2) |
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Linear Regression Model for the Noise |
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343 | (2) |
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Equivalence of ML and Least-Squares Solutions |
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345 | (1) |
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346 | (1) |
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347 | (1) |
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``Significant'' Factors; The R-Statistic |
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347 | (2) |
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Example: Was T2 an Insignificant Factor? |
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349 | (1) |
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Accuracy of the Estimated Coefficients |
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350 | (13) |
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Absorptance of an Optical Fiber |
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350 | (1) |
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Variance of Error in the General Case |
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351 | (2) |
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Error in the Estimated Absorptance of an Optical Fiber |
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353 | (10) |
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Principal Components Analysis |
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363 | (12) |
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363 | (1) |
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Equivalent Eigenvalue Problem |
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364 | (2) |
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The Eigenvalues as Sample Variances |
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366 | (1) |
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The Data in Terms of Principal Components |
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366 | (1) |
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Reduction in Data Dimensionality |
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367 | (1) |
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Return to the H-D Problem |
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368 | (1) |
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Application to Multispectral Imagery |
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368 | (3) |
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371 | (4) |
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The Controversy Between Bayesians and Classicists |
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375 | (12) |
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Bayesian Approach to Confidence Limits for an Estimated Probability |
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376 | (4) |
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Probability Law for the Unknown Probability |
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377 | (1) |
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Assumption of a Uniform Prior |
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377 | |
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Irrelevance of Choice of Prior Statistic p0(x) if N is Large |
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376 | (3) |
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Limiting Form for N Large |
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379 | (1) |
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379 | (1) |
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Laplace's Rule of Succession |
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380 | (7) |
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380 | (3) |
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383 | (1) |
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384 | (3) |
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Introduction to Estimation Methods |
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387 | (64) |
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Deterministic Parameters: Likelihood Theory |
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388 | (10) |
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388 | (1) |
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Maximum Likelihood Estimators |
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389 | (2) |
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Cramer-Rao Lower Bound on Error |
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391 | (2) |
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Achieving the Lower Bound |
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393 | (1) |
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Testing for Efficient Estimators |
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394 | (1) |
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Can a Bound to the Error be Known if an Efficient Estimator Does Not Exist? |
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395 | (2) |
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When can the Bhattacharyya Bound be Achieved? |
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397 | (1) |
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Random Parameters: Bayesian Estimation Theory |
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398 | (15) |
|
|
|
399 | (1) |
|
|
|
400 | (5) |
|
|
|
405 | (8) |
|
Exact Estimates of Probability Laws: The Principle of Extreme Physical Information |
|
|
413 | (38) |
|
A Knowledge-Based View of Nature |
|
|
415 | (1) |
|
Fisher Information as a Bridge Between Noumenon and Phenomenon |
|
|
416 | (3) |
|
|
|
419 | (1) |
|
Invariance of the Fisher Information Length to Unitary Transformation |
|
|
420 | (1) |
|
Multidimensional Form of I |
|
|
421 | (1) |
|
Lorentz Transformation of Special Relativity |
|
|
422 | (1) |
|
What Constants Should be Regarded as Universal Physical Constants? |
|
|
423 | (1) |
|
Transition to Complex Probability Amplitudes |
|
|
424 | (1) |
|
Space-Time Measurement: Information Capacity in Fourier Space |
|
|
424 | (1) |
|
Relation Among Energy, Mass and Momentum |
|
|
425 | (1) |
|
|
|
426 | (1) |
|
Bound Information J and Efficiency Constant k |
|
|
426 | (1) |
|
Perturbing Effect of the Probe Particle |
|
|
427 | (1) |
|
Equality of the Perturbed Informations |
|
|
428 | (3) |
|
EPI Variational Principle, and Framework |
|
|
431 | (2) |
|
The Measurement Process in Detail |
|
|
433 | (2) |
|
|
|
435 | (1) |
|
Free-Field Klein-Gordon Equation |
|
|
435 | (1) |
|
|
|
436 | (1) |
|
Schroedinger Wave Equation (SWE) |
|
|
437 | (1) |
|
Dimensionality, and Plato's Cave |
|
|
438 | (1) |
|
Wheeler's ``Participatory Universe'' |
|
|
439 | (1) |
|
Exhaustivity Property, and Future Research |
|
|
439 | (1) |
|
Can EPI be Used in a Design Mode? |
|
|
440 | (1) |
|
EPI as a Knowledge Acquisition Game |
|
|
440 | (1) |
|
|
|
441 | (10) |
| Appendix |
|
451 | (18) |
|
|
|
451 | (2) |
|
|
|
453 | (2) |
|
|
|
455 | (1) |
|
|
|
456 | (3) |
|
|
|
459 | (1) |
|
|
|
460 | (4) |
|
|
|
464 | (5) |
| References |
|
469 | (10) |
| Index |
|
479 | |
