Tercera Evaluacion
Adriana Velosa - John Mauricio Freyre
1. Dadas las siguientes matrices de transicion de un paso de una cadena de Markov, determine las clases y clasifique los estados. Encuentre las matrices de transicion de varios pasos para ver si converge o no.
a)
\[\left( \begin{array}{cccc} 0 & 0 & 1/3 & 2/3 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \end{array} \right)\]
library(expm)## Loading required package: Matrix##
## Attaching package: 'expm'## The following object is masked from 'package:Matrix':
##
## expmm=matrix(c(0, 1, 0, 0, 0, 0, 1, 1, 1/3, 0, 0, 0, 2/3, 0, 0, 0), nrow = 4)
for (i in 1:10) {
show(m%^%i)
}## [,1] [,2] [,3] [,4]
## [1,] 0 0 0.3333333 0.6666667
## [2,] 1 0 0.0000000 0.0000000
## [3,] 0 1 0.0000000 0.0000000
## [4,] 0 1 0.0000000 0.0000000
## [,1] [,2] [,3] [,4]
## [1,] 0 1 0.0000000 0.0000000
## [2,] 0 0 0.3333333 0.6666667
## [3,] 1 0 0.0000000 0.0000000
## [4,] 1 0 0.0000000 0.0000000
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0.0000000 0.0000000
## [2,] 0 1 0.0000000 0.0000000
## [3,] 0 0 0.3333333 0.6666667
## [4,] 0 0 0.3333333 0.6666667
## [,1] [,2] [,3] [,4]
## [1,] 0 0 0.3333333 0.6666667
## [2,] 1 0 0.0000000 0.0000000
## [3,] 0 1 0.0000000 0.0000000
## [4,] 0 1 0.0000000 0.0000000
## [,1] [,2] [,3] [,4]
## [1,] 0 1 0.0000000 0.0000000
## [2,] 0 0 0.3333333 0.6666667
## [3,] 1 0 0.0000000 0.0000000
## [4,] 1 0 0.0000000 0.0000000
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0.0000000 0.0000000
## [2,] 0 1 0.0000000 0.0000000
## [3,] 0 0 0.3333333 0.6666667
## [4,] 0 0 0.3333333 0.6666667
## [,1] [,2] [,3] [,4]
## [1,] 0 0 0.3333333 0.6666667
## [2,] 1 0 0.0000000 0.0000000
## [3,] 0 1 0.0000000 0.0000000
## [4,] 0 1 0.0000000 0.0000000
## [,1] [,2] [,3] [,4]
## [1,] 0 1 0.0000000 0.0000000
## [2,] 0 0 0.3333333 0.6666667
## [3,] 1 0 0.0000000 0.0000000
## [4,] 1 0 0.0000000 0.0000000
## [,1] [,2] [,3] [,4]
## [1,] 1 0 0.0000000 0.0000000
## [2,] 0 1 0.0000000 0.0000000
## [3,] 0 0 0.3333333 0.6666667
## [4,] 0 0 0.3333333 0.6666667
## [,1] [,2] [,3] [,4]
## [1,] 0 0 0.3333333 0.6666667
## [2,] 1 0 0.0000000 0.0000000
## [3,] 0 1 0.0000000 0.0000000
## [4,] 0 1 0.0000000 0.0000000-La matriz no converge.
-Los estados tienen periodo 3 (diagonal=Periodo =1+1+0.3333333+0.6666667=3)
-Todos Recurrentes
-Una clase
b)
\[\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1/2 & 1/2 & 0 \\ 0 & 1/2 & 1/2 & 0 \\ 1/2 & 0 & 0 & 1/2 \end{array} \right)\]
m=matrix(c(1, 0, 0, 1/2, 0, 1/2, 1/2, 0, 0, 1/2, 1/2, 0, 0, 0, 0, 1/2), nrow = 4)
for (i in 1:10) {
show(m%^%i)
}## [,1] [,2] [,3] [,4]
## [1,] 1.0 0.0 0.0 0.0
## [2,] 0.0 0.5 0.5 0.0
## [3,] 0.0 0.5 0.5 0.0
## [4,] 0.5 0.0 0.0 0.5
## [,1] [,2] [,3] [,4]
## [1,] 1.00 0.0 0.0 0.00
## [2,] 0.00 0.5 0.5 0.00
## [3,] 0.00 0.5 0.5 0.00
## [4,] 0.75 0.0 0.0 0.25
## [,1] [,2] [,3] [,4]
## [1,] 1.000 0.0 0.0 0.000
## [2,] 0.000 0.5 0.5 0.000
## [3,] 0.000 0.5 0.5 0.000
## [4,] 0.875 0.0 0.0 0.125
## [,1] [,2] [,3] [,4]
## [1,] 1.0000 0.0 0.0 0.0000
## [2,] 0.0000 0.5 0.5 0.0000
## [3,] 0.0000 0.5 0.5 0.0000
## [4,] 0.9375 0.0 0.0 0.0625
## [,1] [,2] [,3] [,4]
## [1,] 1.00000 0.0 0.0 0.00000
## [2,] 0.00000 0.5 0.5 0.00000
## [3,] 0.00000 0.5 0.5 0.00000
## [4,] 0.96875 0.0 0.0 0.03125
## [,1] [,2] [,3] [,4]
## [1,] 1.000000 0.0 0.0 0.000000
## [2,] 0.000000 0.5 0.5 0.000000
## [3,] 0.000000 0.5 0.5 0.000000
## [4,] 0.984375 0.0 0.0 0.015625
## [,1] [,2] [,3] [,4]
## [1,] 1.0000000 0.0 0.0 0.0000000
## [2,] 0.0000000 0.5 0.5 0.0000000
## [3,] 0.0000000 0.5 0.5 0.0000000
## [4,] 0.9921875 0.0 0.0 0.0078125
## [,1] [,2] [,3] [,4]
## [1,] 1.0000000 0.0 0.0 0.00000000
## [2,] 0.0000000 0.5 0.5 0.00000000
## [3,] 0.0000000 0.5 0.5 0.00000000
## [4,] 0.9960938 0.0 0.0 0.00390625
## [,1] [,2] [,3] [,4]
## [1,] 1.0000000 0.0 0.0 0.000000000
## [2,] 0.0000000 0.5 0.5 0.000000000
## [3,] 0.0000000 0.5 0.5 0.000000000
## [4,] 0.9980469 0.0 0.0 0.001953125
## [,1] [,2] [,3] [,4]
## [1,] 1.0000000 0.0 0.0 0.0000000000
## [2,] 0.0000000 0.5 0.5 0.0000000000
## [3,] 0.0000000 0.5 0.5 0.0000000000
## [4,] 0.9990234 0.0 0.0 0.0009765625-La matriz no converge.
-Los estados no tienen periodo
-El estado 0 es absorbente.
-Estados 1 y 2 son recurrentes.
-Estado 3 es transitorio
-Tres clases.
c)
\[\left( \begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 1/2 & 1/2 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array} \right)\]
m=matrix(c(0, 0, 1/2, 0, 0, 0, 1/2, 0, 0, 0, 0, 1, 1, 1, 0, 0), nrow = 4)
for (i in 1:10) {
show(m%^%i)
}## [,1] [,2] [,3] [,4]
## [1,] 0.0 0.0 0 1
## [2,] 0.0 0.0 0 1
## [3,] 0.5 0.5 0 0
## [4,] 0.0 0.0 1 0
## [,1] [,2] [,3] [,4]
## [1,] 0.0 0.0 1 0
## [2,] 0.0 0.0 1 0
## [3,] 0.0 0.0 0 1
## [4,] 0.5 0.5 0 0
## [,1] [,2] [,3] [,4]
## [1,] 0.5 0.5 0 0
## [2,] 0.5 0.5 0 0
## [3,] 0.0 0.0 1 0
## [4,] 0.0 0.0 0 1
## [,1] [,2] [,3] [,4]
## [1,] 0.0 0.0 0 1
## [2,] 0.0 0.0 0 1
## [3,] 0.5 0.5 0 0
## [4,] 0.0 0.0 1 0
## [,1] [,2] [,3] [,4]
## [1,] 0.0 0.0 1 0
## [2,] 0.0 0.0 1 0
## [3,] 0.0 0.0 0 1
## [4,] 0.5 0.5 0 0
## [,1] [,2] [,3] [,4]
## [1,] 0.5 0.5 0 0
## [2,] 0.5 0.5 0 0
## [3,] 0.0 0.0 1 0
## [4,] 0.0 0.0 0 1
## [,1] [,2] [,3] [,4]
## [1,] 0.0 0.0 0 1
## [2,] 0.0 0.0 0 1
## [3,] 0.5 0.5 0 0
## [4,] 0.0 0.0 1 0
## [,1] [,2] [,3] [,4]
## [1,] 0.0 0.0 1 0
## [2,] 0.0 0.0 1 0
## [3,] 0.0 0.0 0 1
## [4,] 0.5 0.5 0 0
## [,1] [,2] [,3] [,4]
## [1,] 0.5 0.5 0 0
## [2,] 0.5 0.5 0 0
## [3,] 0.0 0.0 1 0
## [4,] 0.0 0.0 0 1
## [,1] [,2] [,3] [,4]
## [1,] 0.0 0.0 0 1
## [2,] 0.0 0.0 0 1
## [3,] 0.5 0.5 0 0
## [4,] 0.0 0.0 1 0-La matriz no converge.
-Los estados tienen periodo 3
-Todos los estados son recurrentes.
-Una clase.
d)
\[\left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{array} \right)\]
m=matrix(c(0, 0, 1, 1, 0, 0, 0, 1, 0), nrow = 3)
for (i in 1:10) {
show(m%^%i)
}## [,1] [,2] [,3]
## [1,] 0 1 0
## [2,] 0 0 1
## [3,] 1 0 0
## [,1] [,2] [,3]
## [1,] 0 0 1
## [2,] 1 0 0
## [3,] 0 1 0
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1
## [,1] [,2] [,3]
## [1,] 0 1 0
## [2,] 0 0 1
## [3,] 1 0 0
## [,1] [,2] [,3]
## [1,] 0 0 1
## [2,] 1 0 0
## [3,] 0 1 0
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1
## [,1] [,2] [,3]
## [1,] 0 1 0
## [2,] 0 0 1
## [3,] 1 0 0
## [,1] [,2] [,3]
## [1,] 0 0 1
## [2,] 1 0 0
## [3,] 0 1 0
## [,1] [,2] [,3]
## [1,] 1 0 0
## [2,] 0 1 0
## [3,] 0 0 1
## [,1] [,2] [,3]
## [1,] 0 1 0
## [2,] 0 0 1
## [3,] 1 0 0-La matriz no converge.
-Los estados tienen periodo 3
-Todos los estados son recurrentes.
-Una clase.
e)
\[\left( \begin{array}{cccc} 0 & 1/3 & 1/3 & 1/3 \\ 1/3 & 0 & 1/3 & 1/3 \\ 1/3 & 1/3 & 0 & 1/3 \\ 1/3 & 1/3 & 1/3 & 0 \end{array} \right)\]
m=matrix(c(0, 1/3, 1/3,1/3, 1/3, 0, 1/3, 1/3, 1/3, 1/3, 0, 1/3, 1/3, 1/3, 1/3, 0), nrow = 4)
for (i in 10:20) {
show(m%^%i)
}## [,1] [,2] [,3] [,4]
## [1,] 0.2500127 0.2499958 0.2499958 0.2499958
## [2,] 0.2499958 0.2500127 0.2499958 0.2499958
## [3,] 0.2499958 0.2499958 0.2500127 0.2499958
## [4,] 0.2499958 0.2499958 0.2499958 0.2500127
## [,1] [,2] [,3] [,4]
## [1,] 0.2499958 0.2500014 0.2500014 0.2500014
## [2,] 0.2500014 0.2499958 0.2500014 0.2500014
## [3,] 0.2500014 0.2500014 0.2499958 0.2500014
## [4,] 0.2500014 0.2500014 0.2500014 0.2499958
## [,1] [,2] [,3] [,4]
## [1,] 0.2500014 0.2499995 0.2499995 0.2499995
## [2,] 0.2499995 0.2500014 0.2499995 0.2499995
## [3,] 0.2499995 0.2499995 0.2500014 0.2499995
## [4,] 0.2499995 0.2499995 0.2499995 0.2500014
## [,1] [,2] [,3] [,4]
## [1,] 0.2499995 0.2500002 0.2500002 0.2500002
## [2,] 0.2500002 0.2499995 0.2500002 0.2500002
## [3,] 0.2500002 0.2500002 0.2499995 0.2500002
## [4,] 0.2500002 0.2500002 0.2500002 0.2499995
## [,1] [,2] [,3] [,4]
## [1,] 0.2500002 0.2499999 0.2499999 0.2499999
## [2,] 0.2499999 0.2500002 0.2499999 0.2499999
## [3,] 0.2499999 0.2499999 0.2500002 0.2499999
## [4,] 0.2499999 0.2499999 0.2499999 0.2500002
## [,1] [,2] [,3] [,4]
## [1,] 0.2499999 0.2500000 0.2500000 0.2500000
## [2,] 0.2500000 0.2499999 0.2500000 0.2500000
## [3,] 0.2500000 0.2500000 0.2499999 0.2500000
## [4,] 0.2500000 0.2500000 0.2500000 0.2499999
## [,1] [,2] [,3] [,4]
## [1,] 0.25 0.25 0.25 0.25
## [2,] 0.25 0.25 0.25 0.25
## [3,] 0.25 0.25 0.25 0.25
## [4,] 0.25 0.25 0.25 0.25
## [,1] [,2] [,3] [,4]
## [1,] 0.25 0.25 0.25 0.25
## [2,] 0.25 0.25 0.25 0.25
## [3,] 0.25 0.25 0.25 0.25
## [4,] 0.25 0.25 0.25 0.25
## [,1] [,2] [,3] [,4]
## [1,] 0.25 0.25 0.25 0.25
## [2,] 0.25 0.25 0.25 0.25
## [3,] 0.25 0.25 0.25 0.25
## [4,] 0.25 0.25 0.25 0.25
## [,1] [,2] [,3] [,4]
## [1,] 0.25 0.25 0.25 0.25
## [2,] 0.25 0.25 0.25 0.25
## [3,] 0.25 0.25 0.25 0.25
## [4,] 0.25 0.25 0.25 0.25
## [,1] [,2] [,3] [,4]
## [1,] 0.25 0.25 0.25 0.25
## [2,] 0.25 0.25 0.25 0.25
## [3,] 0.25 0.25 0.25 0.25
## [4,] 0.25 0.25 0.25 0.25-La matriz converge.
-No tiene periodico.
-Todos los estados son recurrentes.
-Una clase.
f)
\[\left( \begin{array}{ccc} 0 & 0 & 1 \\ 1/2 & 1/2 & 1 \\ 0 & 1 & 0 \end{array} \right)\]
m=matrix(c(0, 1/2, 0, 0, 1/2, 1, 1, 0, 0), nrow = 3)
for (i in 50:60) {
show(m%^%i)
}## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25
## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25
## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25
## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25
## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25
## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25
## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25
## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25
## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25
## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25
## [,1] [,2] [,3]
## [1,] 0.25 0.5 0.25
## [2,] 0.25 0.5 0.25
## [3,] 0.25 0.5 0.25-La matriz converge.
-No periodico.
-Todos los estados son recurrentes.
-Una clase.
g)
\[\left( \begin{array}{ccccc} 1/4 & 3/4 & 0 & 0 & 0 \\ 3/4 & 1/4 & 0 & 0 & 0 \\ 1/3 & 1/3 & 1/3 & 0 & 0 \\ 0 & 0 & 0 &3/4 & 1/4 \\ 0 & 0 & 0 &1/4 & 3/4 \end{array} \right)\]
m=matrix(c(1/4, 3/4, 1/3, 0, 0, 3/4, 1/4, 1/3, 0, 0, 0, 0, 1/3, 0, 0,0,0,0,3/4,1/4,0,0,0,1/4,3/4), nrow = 5)
for (i in 1:10) {
show(m%^%i)
}## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2500000 0.7500000 0.0000000 0.00 0.00
## [2,] 0.7500000 0.2500000 0.0000000 0.00 0.00
## [3,] 0.3333333 0.3333333 0.3333333 0.00 0.00
## [4,] 0.0000000 0.0000000 0.0000000 0.75 0.25
## [5,] 0.0000000 0.0000000 0.0000000 0.25 0.75
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.6250000 0.3750000 0.0000000 0.000 0.000
## [2,] 0.3750000 0.6250000 0.0000000 0.000 0.000
## [3,] 0.4444444 0.4444444 0.1111111 0.000 0.000
## [4,] 0.0000000 0.0000000 0.0000000 0.625 0.375
## [5,] 0.0000000 0.0000000 0.0000000 0.375 0.625
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.4375000 0.5625000 0.00000000 0.0000 0.0000
## [2,] 0.5625000 0.4375000 0.00000000 0.0000 0.0000
## [3,] 0.4814815 0.4814815 0.03703704 0.0000 0.0000
## [4,] 0.0000000 0.0000000 0.00000000 0.5625 0.4375
## [5,] 0.0000000 0.0000000 0.00000000 0.4375 0.5625
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.5312500 0.4687500 0.00000000 0.00000 0.00000
## [2,] 0.4687500 0.5312500 0.00000000 0.00000 0.00000
## [3,] 0.4938272 0.4938272 0.01234568 0.00000 0.00000
## [4,] 0.0000000 0.0000000 0.00000000 0.53125 0.46875
## [5,] 0.0000000 0.0000000 0.00000000 0.46875 0.53125
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.4843750 0.5156250 0.000000000 0.000000 0.000000
## [2,] 0.5156250 0.4843750 0.000000000 0.000000 0.000000
## [3,] 0.4979424 0.4979424 0.004115226 0.000000 0.000000
## [4,] 0.0000000 0.0000000 0.000000000 0.515625 0.484375
## [5,] 0.0000000 0.0000000 0.000000000 0.484375 0.515625
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.5078125 0.4921875 0.000000000 0.0000000 0.0000000
## [2,] 0.4921875 0.5078125 0.000000000 0.0000000 0.0000000
## [3,] 0.4993141 0.4993141 0.001371742 0.0000000 0.0000000
## [4,] 0.0000000 0.0000000 0.000000000 0.5078125 0.4921875
## [5,] 0.0000000 0.0000000 0.000000000 0.4921875 0.5078125
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.4960938 0.5039062 0.0000000000 0.0000000 0.0000000
## [2,] 0.5039062 0.4960938 0.0000000000 0.0000000 0.0000000
## [3,] 0.4997714 0.4997714 0.0004572474 0.0000000 0.0000000
## [4,] 0.0000000 0.0000000 0.0000000000 0.5039062 0.4960938
## [5,] 0.0000000 0.0000000 0.0000000000 0.4960938 0.5039062
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.5019531 0.4980469 0.0000000000 0.0000000 0.0000000
## [2,] 0.4980469 0.5019531 0.0000000000 0.0000000 0.0000000
## [3,] 0.4999238 0.4999238 0.0001524158 0.0000000 0.0000000
## [4,] 0.0000000 0.0000000 0.0000000000 0.5019531 0.4980469
## [5,] 0.0000000 0.0000000 0.0000000000 0.4980469 0.5019531
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.4990234 0.5009766 0.000000e+00 0.0000000 0.0000000
## [2,] 0.5009766 0.4990234 0.000000e+00 0.0000000 0.0000000
## [3,] 0.4999746 0.4999746 5.080526e-05 0.0000000 0.0000000
## [4,] 0.0000000 0.0000000 0.000000e+00 0.5009766 0.4990234
## [5,] 0.0000000 0.0000000 0.000000e+00 0.4990234 0.5009766
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.5004883 0.4995117 0.000000e+00 0.0000000 0.0000000
## [2,] 0.4995117 0.5004883 0.000000e+00 0.0000000 0.0000000
## [3,] 0.4999915 0.4999915 1.693509e-05 0.0000000 0.0000000
## [4,] 0.0000000 0.0000000 0.000000e+00 0.5004883 0.4995117
## [5,] 0.0000000 0.0000000 0.000000e+00 0.4995117 0.5004883-La matriz converge.
-No periodico.
-Los estados 0,3,4 son recurrentes y el estado 2 es transitorio.
-Tres clases.
2. Determine el periodo de los estados en la cadena de Markov que tiene las siguientes matrices de transicion de un paso:
a)
m=matrix(c(0, 0, 1, 0, 0, 0, 0, 0, 0, 1/4, 0, 1/2, 0, 1, 0, 0, 1, 0, 2/3,
0, 0, 0, 0, 0, 0, 0, 0, 3/4, 0, 1/2, 1/3, 0, 0, 0, 0, 0), nrow = 6)
for (i in 1:10) {
show(m%^%i)
}## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0 0.00 0 0.6666667 0.00 0.3333333
## [2,] 0 0.00 1 0.0000000 0.00 0.0000000
## [3,] 1 0.00 0 0.0000000 0.00 0.0000000
## [4,] 0 0.25 0 0.0000000 0.75 0.0000000
## [5,] 0 0.00 1 0.0000000 0.00 0.0000000
## [6,] 0 0.50 0 0.0000000 0.50 0.0000000
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [2,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [3,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [4,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [5,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [6,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [2,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [3,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [4,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [5,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [6,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [2,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [3,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [4,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [5,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [6,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [2,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [3,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [4,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [5,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [6,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [2,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [3,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [4,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [5,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [6,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [2,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [3,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [4,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [5,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [6,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [2,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [3,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [4,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [5,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [6,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [2,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [3,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [4,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [5,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [6,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [,1] [,2] [,3] [,4] [,5] [,6]
## [1,] 0 0.3333333 0 0.0000000 0.6666667 0.0000000
## [2,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [3,] 0 0.0000000 0 0.6666667 0.0000000 0.3333333
## [4,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000
## [5,] 1 0.0000000 0 0.0000000 0.0000000 0.0000000
## [6,] 0 0.0000000 1 0.0000000 0.0000000 0.0000000El periodo de todos los estados es 4 ya que por su diagonal diferente de cero:
1+0.3333333+1+0.6666667+0.6666667+0.3333333## [1] 4b)
m=matrix(c(0, 1/4, 0, 0, 1/3, 4/5, 0, 1/2, 0, 0, 0, 1/2, 0, 0, 1/3, 1/5,
1/4, 1/10, 1, 1/3, 0, 0, 2/5, 0, 0), nrow = 5)
for (i in 1:10) {
show(m%^%i)
}## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0000000 0.8 0.0000000 0.2000000 0.0
## [2,] 0.2500000 0.0 0.5000000 0.2500000 0.0
## [3,] 0.0000000 0.5 0.0000000 0.1000000 0.4
## [4,] 0.0000000 0.0 0.0000000 1.0000000 0.0
## [5,] 0.3333333 0.0 0.3333333 0.3333333 0.0
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2000000 0.0000000 0.4000000 0.4000000 0.0000000
## [2,] 0.0000000 0.4500000 0.0000000 0.3500000 0.2000000
## [3,] 0.2583333 0.0000000 0.3833333 0.3583333 0.0000000
## [4,] 0.0000000 0.0000000 0.0000000 1.0000000 0.0000000
## [5,] 0.0000000 0.4333333 0.0000000 0.4333333 0.1333333
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0000000 0.3600000 0.0000000 0.4800000 0.1600000
## [2,] 0.1791667 0.0000000 0.2916667 0.5291667 0.0000000
## [3,] 0.0000000 0.3983333 0.0000000 0.4483333 0.1533333
## [4,] 0.0000000 0.0000000 0.0000000 1.0000000 0.0000000
## [5,] 0.1527778 0.0000000 0.2611111 0.5861111 0.0000000
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.1433333 0.0000000 0.2333333 0.6233333 0.0000000
## [2,] 0.0000000 0.2891667 0.0000000 0.5941667 0.1166667
## [3,] 0.1506944 0.0000000 0.2502778 0.5990278 0.0000000
## [4,] 0.0000000 0.0000000 0.0000000 1.0000000 0.0000000
## [5,] 0.0000000 0.2527778 0.0000000 0.6427778 0.1044444
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.00000000 0.2313333 0.0000000 0.6753333 0.09333333
## [2,] 0.11118056 0.0000000 0.1834722 0.7053472 0.00000000
## [3,] 0.00000000 0.2456944 0.0000000 0.6541944 0.10011111
## [4,] 0.00000000 0.0000000 0.0000000 1.0000000 0.00000000
## [5,] 0.09800926 0.0000000 0.1612037 0.7407870 0.00000000
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.08894444 0.0000000 0.1467778 0.7642778 0.00000000
## [2,] 0.00000000 0.1806806 0.0000000 0.7459306 0.07338889
## [3,] 0.09479398 0.0000000 0.1562176 0.7489884 0.00000000
## [4,] 0.00000000 0.0000000 0.0000000 1.0000000 0.00000000
## [5,] 0.00000000 0.1590093 0.0000000 0.7765093 0.06448148
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.00000000 0.1445444 0.0000000 0.7967444 0.05871111
## [2,] 0.06963310 0.0000000 0.1148032 0.8155637 0.00000000
## [3,] 0.00000000 0.1539440 0.0000000 0.7835690 0.06248704
## [4,] 0.00000000 0.0000000 0.0000000 1.0000000 0.00000000
## [5,] 0.06124614 0.0000000 0.1009985 0.8377554 0.00000000
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.05570648 0.00000000 0.09184259 0.8524509 0.00000000
## [2,] 0.00000000 0.11310810 0.00000000 0.8409706 0.04592130
## [3,] 0.05931501 0.00000000 0.09780100 0.8428840 0.00000000
## [4,] 0.00000000 0.00000000 0.00000000 1.0000000 0.00000000
## [5,] 0.00000000 0.09949614 0.00000000 0.8601045 0.04039938
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.00000000 0.09048648 0.00000000 0.8727765 0.03673704
## [2,] 0.04358412 0.00000000 0.07186115 0.8845547 0.00000000
## [3,] 0.00000000 0.09635251 0.00000000 0.8645271 0.03912040
## [4,] 0.00000000 0.00000000 0.00000000 1.0000000 0.00000000
## [5,] 0.03834050 0.00000000 0.06321453 0.8984450 0.00000000
## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.03486730 0.00000000 0.05748892 0.9076438 0.00000000
## [2,] 0.00000000 0.07079787 0.00000000 0.9004577 0.02874446
## [3,] 0.03712826 0.00000000 0.06121639 0.9016554 0.00000000
## [4,] 0.00000000 0.00000000 0.00000000 1.0000000 0.00000000
## [5,] 0.00000000 0.06227966 0.00000000 0.9124345 0.02528581El estado 3 tiene periodo 1, los otros estados no tienen periodo.
3. Suponga que la probabilidad de que llueva mañana es 0.5 si llueve hoy y que la probabilidad de que no llueva mañana es 0.9 si hoy no llueve. Suponga, ademas, que estas probabilidades no cambian. Formule la evolucion del clima como una cadena de Markov, de su matriz de transicion de un paso y haga el diagrama de transicion de estados. Obtenga las probabilidades de estado estable.
m=matrix(c(0.5, 0.1, 0.5, 0.9), nrow = 2)
show(m)## [,1] [,2]
## [1,] 0.5 0.5
## [2,] 0.1 0.9
Con:
\[\pi0 + \pi1=1\]
\[\pi0 =0.5\pi0 +0.1 \pi1 \]
\[\pi1 =0.5\pi0+0.9\pi1 \]
Resolviendo:
A=matrix(c(1, -0.5, 1, 0.1), nrow = 2)
show(A)## [,1] [,2]
## [1,] 1.0 1.0
## [2,] -0.5 0.1B=matrix(c(1, 0), nrow = 2)
show(B)## [,1]
## [1,] 1
## [2,] 0Propabilidades de estado estable:
show(solve(A,B))## [,1]
## [1,] 0.1666667
## [2,] 0.83333334. En el problema anterior substituya 0.5 por \[\alpha\] y 0.9 por \[\beta\] y halle las probabilidades de estado estable en terminos de \[\alpha\] y \[\beta\].
\[\pi0 + \pi1=1\]
\[\pi0 =\alpha \pi0 + (1-\beta)\pi1 \]
\[\pi1 =\alpha \pi0+\beta\pi1 \]
Donde:
\[\pi1 =1-\pi0 \]
y
\[\pi0 =1-\pi1 \]
Reemplazando:
\[\pi0 =\alpha \pi0 + (1-\beta)\pi1 \] \[\pi0 =\alpha \pi0 + (1-\beta) (1-\pi0) \] \[\pi0 =\alpha \pi0 + (1-\pi0-\beta+\beta\pi0) \] \[\pi0 =\alpha \pi0 + 1-\pi0-\beta+\beta\pi0 \] \[\pi0 = \pi0(\alpha -1+ \beta)+ 1- \beta \] \[\pi0 - \pi0(\alpha -1+ \beta) = 1- \beta \] \[\pi0 (1- (\alpha -1+ \beta)) = 1- \beta \] \[\pi0 (1- \alpha +1- \beta) = 1- \beta \] \[\pi0 (2- \alpha - \beta) = 1- \beta \] \[\pi0 = \frac{1- \beta}{(2- \alpha - \beta)} \]
Reemplazando:
\[\pi1 =\alpha \pi0+\beta\pi1 \] \[\pi1 =\alpha (1-\pi1)+\beta\pi1 \] \[\pi1 =\alpha -\pi1\alpha+\beta\pi1 \] \[\pi1+ \pi1\alpha -\beta\pi1 =\alpha \] \[\pi1 (1+\alpha -\beta) =\alpha \] \[\pi1 = \frac{\alpha}{1+\alpha -\beta} \]
5. La cerveceria A esta preocupada por su mayor competidor B. Suponga que el cambio de marca se puede modelar como una cadena de markov incluyendo tres estados: A y B representan los clientes de las mencionadas cervezas y el estado C representa las demas marcas. Los datos se toman cada mes y se construye la siguiente matriz de transicion de un paso con datos historicos.
Haga el diagrama de transicion de estados. ¿Cuales son los porcentajes de participacion del mercado de cervezas?
Resolviendo:
A=matrix(c(0.7, 0.2, 0.1, 0.2, 0.75, 0.1, 0.1, 0.05, 0.8), nrow = 3)
show(A%^%60)## [,1] [,2] [,3]
## [1,] 0.3461538 0.3846154 0.2692308
## [2,] 0.3461538 0.3846154 0.2692308
## [3,] 0.3461538 0.3846154 0.2692308Propabilidades de estado estable:
A= 0.3461538
B= 0.3846154
C=0.2692308
Los porcentajes de participacion del mercado de cervezas son:
A= 34.61538 %
B= 38.46154 %
C=26.92308 %
6. Suponga que una red de comunicaciones transmite digitos binarios (0,1), en donde cada digito se transmite un numero indefinido de veces. Durante cada transmision, la probabilidad de que ese digito se transmita correctamente es de 0.99. En otras palabras, se tiene una probabilidad de 0.01 de que el digito transmitido se registre con el valor opuesto al final de la transmision.Determine la matriz de un paso. Resuelva las ecuaciones de estado estable. ¿Que sucede si se rediseña la red para mejorar la probabilidad de la exactitud de una transmision de 0.99 a 0.999.
Matriz de un paso:
m=matrix(c(0.99, 0.01, 0.01, 0.99), nrow = 2)
show(m)## [,1] [,2]
## [1,] 0.99 0.01
## [2,] 0.01 0.99Teniendo en cuenta que:
\[\pi0 + \pi1=1\]
\[\pi0 =0.99\pi0 + 0.01\pi1 \]
\[\pi1 =0.01 \pi0+ 0.99\pi1 \]
Con:
A=matrix(c(1, -0.01, 1, 0.01), nrow = 2)
show(A)## [,1] [,2]
## [1,] 1.00 1.00
## [2,] -0.01 0.01B=matrix(c(1, 0), nrow = 2)
show(B)## [,1]
## [1,] 1
## [2,] 0Las probabilidades de estado estable:
show(solve(A,B))## [,1]
## [1,] 0.5
## [2,] 0.5Con probabilidad de 0.999 y 0.001
A=matrix(c(1, -0.001, 1, 0.001), nrow = 2)
show(A)## [,1] [,2]
## [1,] 1.000 1.000
## [2,] -0.001 0.001B=matrix(c(1, 0), nrow = 2)
show(B)## [,1]
## [1,] 1
## [2,] 0Las probabilidades de estado estable:
show(solve(A,B))## [,1]
## [1,] 0.5
## [2,] 0.57. Un proceso de produccion incluye una maquina que se deteriora con rapidez, tanto en cantidad como en calidad de produccion con el trabajo pesado, por lo que se inspecciona al final de cada dıa. despues de la inspeccion se clasifica la condici´on de la m´aquina en uno de cuatro estados:
0: tan buena como nueva 1: operable con deterioro m´ınimo 2: operable con deterioro mayor 3: inoperable y reemplazada por nueva
El proceso se puede modelar como una cadena de Markov con matriz de transicion de un paso dada por:
a) Encuentre las probabilidades de estado estable.
m=matrix(c(0, 0, 0, 1, 7/8, 3/4, 0, 0, 1/16, 1/8, 1/2, 0, 1/16, 1/8, 1/2,
0), nrow = 4)
show(m%^%60)## [,1] [,2] [,3] [,4]
## [1,] 0.1538462 0.5384615 0.1538462 0.1538462
## [2,] 0.1538462 0.5384615 0.1538462 0.1538462
## [3,] 0.1538462 0.5384615 0.1538462 0.1538462
## [4,] 0.1538462 0.5384615 0.1538462 0.1538462Las probabilidades de estado estable:
\[\pi0 =0.1538462\] \[\pi1= 0.5384615\] \[\pi2= 0.1538462\] \[\pi3= 0.1538462\]
b) Si los costos respectivos por estar en los estados 0,1,2 y 3 son $0, $1000, $3000 y $6000,¿Cual es el costo diario esperado a la larga?
Costo diario esperado:
((7/8) * 1000) + ((1/16) * 3000) + ((1/16) * 6000) + ((3/4) * 1000) +
((1/8) * 3000) + ((1/8) * 6000) + ((1/2) * 3000) + ((1/2) * 6000) + 6000## [1] 13812.58. Una particula se mueve sobre un circulo por puntos marcados 0,1,2,3 y 4 que estan en el sentido de las agujas del reloj. En cada paso la particula tiene una probabilidad de 0.5 de moverse en el sentido de las agujas del reloj y de 0.5 de moverse en sentido opuesto. Sea Xn la localizacion de la particula despues de n pasos.
a) Encuentre la matriz de transicion de un paso.
matrix(c(0, 0.5, 0, 0, 0.5, 0.5, 0, 0.5, 0, 0, 0, 0.5, 0, 0.5, 0, 0, 0,
0.5, 0, 0.5, 0.5, 0, 0, 0.5, 0), nrow = 5)## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.0 0.5 0.0 0.0 0.5
## [2,] 0.5 0.0 0.5 0.0 0.0
## [3,] 0.0 0.5 0.0 0.5 0.0
## [4,] 0.0 0.0 0.5 0.0 0.5
## [5,] 0.5 0.0 0.0 0.5 0.0b) Discuta a cerca de la convergencia de la matriz.
La matriz es convergente.
m=matrix(c(0, 0.5, 0, 0, 0.5, 0.5, 0, 0.5, 0, 0, 0, 0.5, 0, 0.5, 0, 0, 0,
0.5, 0, 0.5, 0.5, 0, 0, 0.5, 0), nrow = 5)
show(m%^%20)## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2057705 0.1953316 0.2017832 0.2017832 0.1953316
## [2,] 0.1953316 0.2057705 0.1953316 0.2017832 0.2017832
## [3,] 0.2017832 0.1953316 0.2057705 0.1953316 0.2017832
## [4,] 0.2017832 0.2017832 0.1953316 0.2057705 0.1953316
## [5,] 0.1953316 0.2017832 0.2017832 0.1953316 0.2057705m=matrix(c(0, 0.5, 0, 0, 0.5, 0.5, 0, 0.5, 0, 0, 0, 0.5, 0, 0.5, 0, 0, 0,
0.5, 0, 0.5, 0.5, 0, 0, 0.5, 0), nrow = 5)
show(m%^%40)## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2000832 0.1999327 0.2000257 0.2000257 0.1999327
## [2,] 0.1999327 0.2000832 0.1999327 0.2000257 0.2000257
## [3,] 0.2000257 0.1999327 0.2000832 0.1999327 0.2000257
## [4,] 0.2000257 0.2000257 0.1999327 0.2000832 0.1999327
## [5,] 0.1999327 0.2000257 0.2000257 0.1999327 0.2000832m=matrix(c(0, 0.5, 0, 0, 0.5, 0.5, 0, 0.5, 0, 0, 0, 0.5, 0, 0.5, 0, 0, 0,
0.5, 0, 0.5, 0.5, 0, 0, 0.5, 0), nrow = 5)
show(m%^%80)## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2 0.2 0.2 0.2 0.2
## [2,] 0.2 0.2 0.2 0.2 0.2
## [3,] 0.2 0.2 0.2 0.2 0.2
## [4,] 0.2 0.2 0.2 0.2 0.2
## [5,] 0.2 0.2 0.2 0.2 0.2m=matrix(c(0, 0.5, 0, 0, 0.5, 0.5, 0, 0.5, 0, 0, 0, 0.5, 0, 0.5, 0, 0, 0,
0.5, 0, 0.5, 0.5, 0, 0, 0.5, 0), nrow = 5)
show(m%^%100)## [,1] [,2] [,3] [,4] [,5]
## [1,] 0.2 0.2 0.2 0.2 0.2
## [2,] 0.2 0.2 0.2 0.2 0.2
## [3,] 0.2 0.2 0.2 0.2 0.2
## [4,] 0.2 0.2 0.2 0.2 0.2
## [5,] 0.2 0.2 0.2 0.2 0.2c) Halle las probabilidades de estado estable.
Las probabilidades de estado estable son 0.2 para todos los estados
\[\pi0 =0.2\] \[\pi1= 0.2\] \[\pi2= 0.2\] \[\pi3= 0.2\]