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1 |
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2 ============================= |
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3 Continuous random variables |
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4 ============================= |
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5 .. contents:: |
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6 :local: |
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7 |
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8 Probability density function |
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9 ============================ |
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10 |
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11 .. role:: def |
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12 :class: def |
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13 |
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14 :def:`Probability density function` (PDF) for continuous random variable |
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15 :math:`x` is function: |
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16 |
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17 .. math:: |
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18 |
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19 PDF(a ≤ X ≤ b) = P(a ≤ X ≤ b) = ∫_{a, b} f_X(x) dx |
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20 |
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21 f_X(x) ≥ 0 |
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22 |
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23 ∫_{-∞, +∞} f_X(x) dx = 1 |
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24 |
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25 :math:`f_X(x)` funtion maps values :math:`x` from sample space to real numbers. |
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26 |
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27 For continuous random variable: |
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28 |
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29 .. math:: P(X = a) = 0 |
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30 |
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31 Expectation |
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32 =========== |
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33 |
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34 :def:`Expectation` of continuous random variable is: |
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35 |
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36 .. math:: μ = E[X] = ∫_{-∞, +∞} x·f_X(x)·dx |
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37 |
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38 Properties: |
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39 |
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40 .. math:: |
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41 |
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42 E[X + Y] = E[X] + E[Y] |
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43 |
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44 E[a·X] = a·E[X] |
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45 |
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46 E[a·X + b] = a·E[X] + b |
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47 |
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48 Variance |
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49 ======== |
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50 |
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51 :def:`Variance` of continuous random variable is: |
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52 |
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53 .. math:: var[X] = ∫_{-∞, +∞} (x-μ)²·f_X(x)·dx |
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54 |
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55 Properties: |
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56 |
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57 .. math:: |
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58 |
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59 var[a·X + b] = a²·var[X] |
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60 |
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61 var[X] = E[X²] - E²[X] |
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62 |
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63 Standard deviation |
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64 ================== |
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65 |
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66 :def:`Standard deviation` of continuous random variable is: |
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67 |
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68 .. math:: σ_Χ = sqrt(var[X]) |
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69 |
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70 |
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71 Continuous uniform random variable |
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72 ================================== |
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73 |
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74 :def:`Continuous uniform random variable` is :math:`f_X(x)` that is non-zero |
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75 only on :math:`[a, b]` with :math:`f_X(x) = `1/(b-a)`. |
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76 |
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77 .. math:: |
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78 |
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79 E[unif(a, b)] = (b+a)/2 |
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80 |
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81 var[unif(a, b)] = (b-a)²/12 |
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82 |
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83 σ = (b-a)/sqrt(12) |
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84 |
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85 Proofs: |
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86 |
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87 .. math:: |
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88 |
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89 E[unif(a, b)] = ∫_{a, b} x·1/(b-a)·dx = x²/2/(b-a) |_{a, b} = (b²-a²)/(b-a)/2 = (b+a)/2 |
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90 |
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91 E[unif²(a, b)] = ∫_{a, b} x²·1/(b-a)·dx = x³/3/(b-a) |_{a, b} = (b³-a³)/(b-a)/3 = (b²+b·a+a²)/3 |
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92 |
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93 var[unif(a, b)] = E[unif²(a, b)] - E²[unif(a, b)] = (b²+b·a+a²)/3 - (b+a)²/4 = (b-a)²/12 |
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94 |
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95 .. note:: |
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96 |
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97 In maxima:: |
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98 |
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99 (%i4) factor((b^2+b*a+a^2)/3 - (a+b)^2/4); |
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100 2 |
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101 (b - a) |
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102 -------- |
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103 12 |
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104 |
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105 Exponential random variables |
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106 ============================ |
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107 |
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108 :def:`Exponential random variables` with parameter :math:`λ` is: |
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109 |
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110 .. math:: f_X(x) = λ·exp(-λ·x) |
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111 |
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112 for :math:`x ≥ 0`, and zero otherwise. |
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113 |
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114 Properties: |
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115 |
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116 .. math:: |
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117 |
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118 E[exp(λ)] = 1/λ |
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119 |
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120 var[exp(λ)] = 1/λ² |
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121 |
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122 Proof: |
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123 |
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124 .. math:: |
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125 |
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126 ∫_{-∞, +∞} f_X(x)·dx = ∫_{0, +∞} λ·exp(-λ·x)·dx = -exp(-λ·x) |_{0, +∞} = 1 |
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127 |
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128 E[exp(λ)] = ∫_{0, +∞} x·λ·exp(-λ·x)·dx = 1/λ |
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129 |
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130 E[exp²(λ)] = ∫_{0, +∞} x²·λ·exp(-λ·x)·dx = 1/λ² |
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131 |
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132 .. note:: |
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133 |
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134 From maxima:: |
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135 |
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136 (%i15) assume(lambda>0); |
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137 (%o15) [lambda > 0] |
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138 |
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139 (%i16) integrate(lambda*%e^(-lambda*x),x,0,inf); |
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140 (%o16) 1 |
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141 |
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142 (%i17) integrate(x*lambda*%e^(-lambda*x),x,0,inf); |
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143 1 |
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144 (%o17) ------ |
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145 lambda |
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146 |
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147 (%i18) integrate(x^2*lambda*%e^(-lambda*x),x,0,inf); |
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148 2 |
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149 (%o18) ------- |
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150 2 |
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151 lambda |
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152 |
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153 Cumulative distribution functions |
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154 ================================= |
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155 |
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156 :def:`Cumulative distribution functions` of random variable :math:`X` is: |
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157 |
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158 .. math:: F_X(x) = P(X ≤ x) = ∫_{-∞, x} f_X(t)·dt |