Tarea 5
Nicolás E. Galeano Sánchez
19/10/2020
Parte teórica del modelo de Poisson(DEMOSTRACIÓN)
Partiremos de la ecuación de Poisson para realizar la demostración:
\[P(x)= \frac{e^-λ*λ^x}{x!}\] Luego Planteamos la distribución teórica:
\[E(x)=\sum_{n=0}^\infty x*P(x)\] Ahora, cuando reemplazamos la ecuación de Poisson en la distribución teórica, obtenemos que: \[E(x)=\sum_{n=0}^\infty \frac{e^-λ*λ^x}{(x-1)!}\]
Factorizmos: \[E(x)=λ\sum_{n=0}^\infty \frac{e^-λ*λ^{x-1}}{(x-1)!}\] Ahora, asumamos que y= x-1 y procedamos a reemplazar: \[E(x)=λ\sum_{n=0}^\infty \frac{e^-λ*λ^y}{y!}\] Por tanto: \[P(y)= \frac{e^-λ*λ^y}{y!}\] En ese sentido: \[E(x)=λ\sum_{n=0}^\infty P(y)\]
La suma de todas las probabilidades que cobija "y2 es igual a 1, por tanto: \[E(x)=λ\]
Parte práctica
Supongamos que X va a ser el número de homicidios reportados por día en CAI de un barrio altamente peligroso en la ciudad de Cali:
x=c(10,8,6,8,4,12,5,11,9,6,15)Ahora calculamos el λ que maximiza nuestra muestra:
calc_verosimil=function(l){
verosimiltud=prod(dpois(x,lambda = l))
return(verosimiltud)
}
lambdas=seq(0,20,0.1)
probas=sapply(lambdas, calc_verosimil)
plot(lambdas,probas,type="l")
De la gráfica vemos que el mayor lambda está al rededor de 10, sin embargo Para ser más precisos creamos un data.frame que determine el mayor λ:
resultados=data.frame(lambdas,probas)
resultados## lambdas probas
## 1 0.0 0.000000e+00
## 2 0.1 4.165424e-154
## 3 0.2 2.746343e-126
## 4 0.3 3.262941e-110
## 5 0.4 6.027352e-99
## 6 0.5 2.581926e-90
## 7 0.6 2.383730e-83
## 8 0.7 1.557871e-77
## 9 0.8 1.465718e-72
## 10 0.9 3.138077e-68
## 11 1.0 2.089988e-64
## 12 1.1 5.411682e-61
## 13 1.2 6.422928e-58
## 14 1.3 3.959562e-55
## 15 1.4 1.397280e-52
## 16 1.5 3.048615e-50
## 17 1.6 4.376013e-48
## 18 1.7 4.347854e-46
## 19 1.8 3.118659e-44
## 20 1.9 1.672888e-42
## 21 2.0 6.913916e-41
## 22 2.1 2.258366e-39
## 23 2.2 5.959209e-38
## 24 2.3 1.294641e-36
## 25 2.4 2.354320e-35
## 26 2.5 3.636183e-34
## 27 2.6 4.831207e-33
## 28 2.7 5.585098e-32
## 29 2.8 5.675028e-31
## 30 2.9 5.114474e-30
## 31 3.0 4.121574e-29
## 32 3.1 2.991855e-28
## 33 3.2 1.969316e-27
## 34 3.3 1.182507e-26
## 35 3.4 6.513099e-26
## 36 3.5 3.307057e-25
## 37 3.6 1.555094e-24
## 38 3.7 6.800796e-24
## 39 3.8 2.776718e-23
## 40 3.9 1.062241e-22
## 41 4.0 3.820015e-22
## 42 4.1 1.295342e-21
## 43 4.2 4.153473e-21
## 44 4.3 1.262667e-20
## 45 4.4 3.648226e-20
## 46 4.5 1.004113e-19
## 47 4.6 2.638266e-19
## 48 4.7 6.630664e-19
## 49 4.8 1.597022e-18
## 50 4.9 3.692683e-18
## 51 5.0 8.210451e-18
## 52 5.1 1.758163e-17
## 53 5.2 3.631224e-17
## 54 5.3 7.243468e-17
## 55 5.4 1.397347e-16
## 56 5.5 2.610116e-16
## 57 5.6 4.726260e-16
## 58 5.7 8.305304e-16
## 59 5.8 1.417838e-15
## 60 5.9 2.353750e-15
## 61 6.0 3.803337e-15
## 62 6.1 5.987257e-15
## 63 6.2 9.190058e-15
## 64 6.3 1.376535e-14
## 65 6.4 2.013581e-14
## 66 6.5 2.878612e-14
## 67 6.6 4.024701e-14
## 68 6.7 5.506951e-14
## 69 6.8 7.378935e-14
## 70 6.9 9.688275e-14
## 71 7.0 1.247165e-13
## 72 7.1 1.574958e-13
## 73 7.2 1.952157e-13
## 74 7.3 2.376211e-13
## 75 7.4 2.841803e-13
## 76 7.5 3.340774e-13
## 77 7.6 3.862265e-13
## 78 7.7 4.393075e-13
## 79 7.8 4.918234e-13
## 80 7.9 5.421747e-13
## 81 8.0 5.887453e-13
## 82 8.1 6.299942e-13
## 83 8.2 6.645428e-13
## 84 8.3 6.912539e-13
## 85 8.4 7.092933e-13
## 86 8.5 7.181713e-13
## 87 8.6 7.177605e-13
## 88 8.7 7.082897e-13
## 89 8.8 6.903168e-13
## 90 8.9 6.646821e-13
## 91 9.0 6.324487e-13
## 92 9.1 5.948348e-13
## 93 9.2 5.531429e-13
## 94 9.3 5.086918e-13
## 95 9.4 4.627559e-13
## 96 9.5 4.165131e-13
## 97 9.6 3.710067e-13
## 98 9.7 3.271184e-13
## 99 9.8 2.855545e-13
## 100 9.9 2.468439e-13
## 101 10.0 2.113441e-13
## 102 10.1 1.792567e-13
## 103 10.2 1.506463e-13
## 104 10.3 1.254635e-13
## 105 10.4 1.035686e-13
## 106 10.5 8.475485e-14
## 107 10.6 6.876982e-14
## 108 10.7 5.533472e-14
## 109 10.8 4.416027e-14
## 110 10.9 3.495953e-14
## 111 11.0 2.745764e-14
## 112 11.1 2.139867e-14
## 113 11.2 1.654996e-14
## 114 11.3 1.270436e-14
## 115 11.4 9.680797e-15
## 116 11.5 7.323660e-15
## 117 11.6 5.501210e-15
## 118 11.7 4.103499e-15
## 119 11.8 3.039961e-15
## 120 11.9 2.236917e-15
## 121 12.0 1.635116e-15
## 122 12.1 1.187442e-15
## 123 12.2 8.568163e-16
## 124 12.3 6.143561e-16
## 125 12.4 4.377783e-16
## 126 12.5 3.100510e-16
## 127 12.6 2.182726e-16
## 128 12.7 1.527544e-16
## 129 12.8 1.062814e-16
## 130 12.9 7.352396e-17
## 131 13.0 5.057633e-17
## 132 13.1 3.459793e-17
## 133 13.2 2.353823e-17
## 134 13.3 1.592776e-17
## 135 13.4 1.072081e-17
## 136 13.5 7.178386e-18
## 137 13.6 4.781741e-18
## 138 13.7 3.169115e-18
## 139 13.8 2.089849e-18
## 140 13.9 1.371349e-18
## 141 14.0 8.955055e-19
## 142 14.1 5.819766e-19
## 143 14.2 3.764345e-19
## 144 14.3 2.423531e-19
## 145 14.4 1.553143e-19
## 146 14.5 9.908444e-20
## 147 14.6 6.293000e-20
## 148 14.7 3.979192e-20
## 149 14.8 2.505202e-20
## 150 14.9 1.570460e-20
## 151 15.0 9.803292e-21
## 152 15.1 6.094004e-21
## 153 15.2 3.772620e-21
## 154 15.3 2.326035e-21
## 155 15.4 1.428386e-21
## 156 15.5 8.736824e-22
## 157 15.6 5.323071e-22
## 158 15.7 3.230676e-22
## 159 15.8 1.953297e-22
## 160 15.9 1.176543e-22
## 161 16.0 7.060455e-23
## 162 16.1 4.221460e-23
## 163 16.2 2.514883e-23
## 164 16.3 1.492854e-23
## 165 16.4 8.830398e-24
## 166 16.5 5.205056e-24
## 167 16.6 3.057532e-24
## 168 16.7 1.789926e-24
## 169 16.8 1.044324e-24
## 170 16.9 6.072803e-25
## 171 17.0 3.519765e-25
## 172 17.1 2.033412e-25
## 173 17.2 1.170958e-25
## 174 17.3 6.721666e-26
## 175 17.4 3.846348e-26
## 176 17.5 2.194178e-26
## 177 17.6 1.247849e-26
## 178 17.7 7.075132e-27
## 179 17.8 3.999482e-27
## 180 17.9 2.254159e-27
## 181 18.0 1.266751e-27
## 182 18.1 7.098030e-28
## 183 18.2 3.965868e-28
## 184 18.3 2.209563e-28
## 185 18.4 1.227596e-28
## 186 18.5 6.801403e-29
## 187 18.6 3.757932e-29
## 188 18.7 2.070710e-29
## 189 18.8 1.137947e-29
## 190 18.9 6.236916e-30
## 191 19.0 3.409376e-30
## 192 19.1 1.858870e-30
## 193 19.2 1.010890e-30
## 194 19.3 5.483422e-31
## 195 19.4 2.966903e-31
## 196 19.5 1.601291e-31
## 197 19.6 8.621117e-32
## 198 19.7 4.630140e-32
## 199 19.8 2.480692e-32
## 200 19.9 1.325898e-32
## 201 20.0 7.069956e-33Obtenemos que el λ= 9.6 con probabilidad de 3.710067e-13 es el que tiene la mayor probabilidad teórica y es consistente con la practica
which.max(resultados$probas)## [1] 86mean(x)## [1] 8.545455