1. Vectores en R

Un vector es una colecciĆ³n ordenada de nĆŗmeros.

x <- c(1,2,3,4,5)
x
## [1] 1 2 3 4 5
y <- c(1,1,1,1,1)
y
## [1] 1 1 1 1 1
y <- rep(1,100)
y
##   [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##  [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##  [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
z <- seq(0,1,length=100)
z
##   [1] 0.00000000 0.01010101 0.02020202 0.03030303 0.04040404 0.05050505
##   [7] 0.06060606 0.07070707 0.08080808 0.09090909 0.10101010 0.11111111
##  [13] 0.12121212 0.13131313 0.14141414 0.15151515 0.16161616 0.17171717
##  [19] 0.18181818 0.19191919 0.20202020 0.21212121 0.22222222 0.23232323
##  [25] 0.24242424 0.25252525 0.26262626 0.27272727 0.28282828 0.29292929
##  [31] 0.30303030 0.31313131 0.32323232 0.33333333 0.34343434 0.35353535
##  [37] 0.36363636 0.37373737 0.38383838 0.39393939 0.40404040 0.41414141
##  [43] 0.42424242 0.43434343 0.44444444 0.45454545 0.46464646 0.47474747
##  [49] 0.48484848 0.49494949 0.50505051 0.51515152 0.52525253 0.53535354
##  [55] 0.54545455 0.55555556 0.56565657 0.57575758 0.58585859 0.59595960
##  [61] 0.60606061 0.61616162 0.62626263 0.63636364 0.64646465 0.65656566
##  [67] 0.66666667 0.67676768 0.68686869 0.69696970 0.70707071 0.71717172
##  [73] 0.72727273 0.73737374 0.74747475 0.75757576 0.76767677 0.77777778
##  [79] 0.78787879 0.79797980 0.80808081 0.81818182 0.82828283 0.83838384
##  [85] 0.84848485 0.85858586 0.86868687 0.87878788 0.88888889 0.89898990
##  [91] 0.90909091 0.91919192 0.92929293 0.93939394 0.94949495 0.95959596
##  [97] 0.96969697 0.97979798 0.98989899 1.00000000
w <- seq(0,1,by=0.01)
w
##   [1] 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14
##  [16] 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
##  [31] 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44
##  [46] 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59
##  [61] 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74
##  [76] 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89
##  [91] 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00

2. Matrices

Una matriz es una estructura bidimensional de datos.

DefiniciĆ³n

Si \(A\) es una matriz de dimensiĆ³n \(m \times n\):

\[ A = \begin{pmatrix} a_{11} & a_{12} & \dots & a_{1n}\\ a_{21} & a_{22} & \dots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ a_{m1} & a_{m2} & \dots & a_{mn} \end{pmatrix} \]

x <- matrix(c(1,2,4,5),2,2)
x
##      [,1] [,2]
## [1,]    1    4
## [2,]    2    5
x <- matrix(c(1,2,4,5),2,2,byrow=TRUE)
x
##      [,1] [,2]
## [1,]    1    2
## [2,]    4    5
y <- matrix(c(2,2,4,4),2,2,byrow=TRUE)
y
##      [,1] [,2]
## [1,]    2    2
## [2,]    4    4

Suma de matrices

Si \(A\) y \(B\) tienen la misma dimensiĆ³n:

\[ C = A + B \]

x + y
##      [,1] [,2]
## [1,]    3    4
## [2,]    8    9

Producto de Hadamard

MultiplicaciĆ³n elemento a elemento:

\[ C_{ij} = A_{ij}B_{ij} \]

x * y
##      [,1] [,2]
## [1,]    2    4
## [2,]   16   20

Producto matricial

\[ C = AB \]

x %*% y
##      [,1] [,2]
## [1,]   10   10
## [2,]   28   28

3. Distribuciones de Probabilidad

En R:

Ejemplo con distribuciĆ³n normal:

\[ X \sim N(\mu, \sigma^2) \]

x <- rnorm(1000,5,4)
hist(x,col="green", main="Normal")

mean(x)
## [1] 5.108045
var(x)
## [1] 15.16915
sd(x)
## [1] 3.894759
sqrt(var(x))
## [1] 3.894759
sd(x) == sqrt(var(x))
## [1] TRUE
plot(density(x),col=4)

4. RegresiĆ³n Lineal

Modelo de regresiĆ³n:

\[ Y = \beta_0 + \beta_1 X + \varepsilon \]

donde:

Modelo con ruido

x <- rnorm(1000)
y <- 3 + 4*x + rnorm(1000)

plot(x,y,col="pink")

fit <- lm(y~x)
fit
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##       3.025        4.009
abline(a=fit$coefficients[1],
       b=fit$coefficients[2],
       col="red",lwd=2)

summary(fit)
## 
## Call:
## lm(formula = y ~ x)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.3554 -0.7019 -0.0198  0.6977  2.9384 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  3.02504    0.03194   94.71   <2e-16 ***
## x            4.00858    0.03291  121.82   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 1.01 on 998 degrees of freedom
## Multiple R-squared:  0.937,  Adjusted R-squared:  0.9369 
## F-statistic: 1.484e+04 on 1 and 998 DF,  p-value: < 2.2e-16

HipĆ³tesis

\[ H_0 : \beta_1 = 0 \]

Se rechaza \(H_0\) si:

\[ p\text{-valor} < \alpha \]

donde:

\[ \alpha = 0.05, 0.01, 0.10 \]

5. SimulaciĆ³n de muestras

GeneraciĆ³n de mĆŗltiples muestras:

n <- 100
M <- 1000

x <- matrix(0,M,n)

for(i in 1:M){
  x[i,] <- rnorm(n,10,5)
}

media <- NULL

for(i in 1:M){
  media[i] <- mean(x[i,])
}

sd(media)
## [1] 0.5152265

Error estƔndar de la media:

\[ SE = \frac{\sigma}{\sqrt{n}} \]

6. Ley de los Grandes NĆŗmeros

Si \(X_1, X_2, ..., X_n\) son i.i.d.:

\[ \bar{X}_n \to \mu \]

cuando \(n \to \infty\).

x <- list()

for(i in 1:1000){
  x[[i]] <- rgamma(i,4)
}

med <- NULL

for(i in 1:1000){
  med[i] <- mean(x[[i]])
}

plot(med,type="l",col="green")
abline(h=4,col="red")

7. Teorema del LĆ­mite Central

Si \(X_1,...,X_n\) son i.i.d. con media \(\mu\) y varianza \(\sigma^2\):

\[ \frac{\bar{X}-\mu}{\sigma/\sqrt{n}} \rightarrow N(0,1) \]

Ejemplo con Gamma

M <- 1000
n <- 100

data <- matrix(0,M,n)

for(i in 1:M){
  data[i,] <- rgamma(n,4)
}

med <- NULL

for(i in 1:M){
  med[i] <- mean(data[i,])
}

hist(med,col="green")

Ejemplo con Cauchy

data <- matrix(0,M,n)

for(i in 1:M){
  data[i,] <- rcauchy(n,4)
}

med <- NULL

for(i in 1:M){
  med[i] <- mean(data[i,])
}

hist(med,col="green")

plot(density(med))

La distribuciĆ³n de Cauchy no tiene momentos finitos, por lo que el TLC no aplica.

Ejemplo con Poisson

data <- matrix(0,M,n)

for(i in 1:M){
  data[i,] <- rpois(n,4)
}

med <- NULL

for(i in 1:M){
  med[i] <- mean(data[i,])
}

hist(med,col="green")

plot(density(med))

x<-c(1,2,3,4,5)
x
## [1] 1 2 3 4 5
y<-c(1,1,1,1,1)
y
## [1] 1 1 1 1 1
y<-rep(1,100)
y
##   [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##  [38] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
##  [75] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
z<-seq(0,1,length=100)
z
##   [1] 0.00000000 0.01010101 0.02020202 0.03030303 0.04040404 0.05050505
##   [7] 0.06060606 0.07070707 0.08080808 0.09090909 0.10101010 0.11111111
##  [13] 0.12121212 0.13131313 0.14141414 0.15151515 0.16161616 0.17171717
##  [19] 0.18181818 0.19191919 0.20202020 0.21212121 0.22222222 0.23232323
##  [25] 0.24242424 0.25252525 0.26262626 0.27272727 0.28282828 0.29292929
##  [31] 0.30303030 0.31313131 0.32323232 0.33333333 0.34343434 0.35353535
##  [37] 0.36363636 0.37373737 0.38383838 0.39393939 0.40404040 0.41414141
##  [43] 0.42424242 0.43434343 0.44444444 0.45454545 0.46464646 0.47474747
##  [49] 0.48484848 0.49494949 0.50505051 0.51515152 0.52525253 0.53535354
##  [55] 0.54545455 0.55555556 0.56565657 0.57575758 0.58585859 0.59595960
##  [61] 0.60606061 0.61616162 0.62626263 0.63636364 0.64646465 0.65656566
##  [67] 0.66666667 0.67676768 0.68686869 0.69696970 0.70707071 0.71717172
##  [73] 0.72727273 0.73737374 0.74747475 0.75757576 0.76767677 0.77777778
##  [79] 0.78787879 0.79797980 0.80808081 0.81818182 0.82828283 0.83838384
##  [85] 0.84848485 0.85858586 0.86868687 0.87878788 0.88888889 0.89898990
##  [91] 0.90909091 0.91919192 0.92929293 0.93939394 0.94949495 0.95959596
##  [97] 0.96969697 0.97979798 0.98989899 1.00000000
w<-seq(0,1,by=0.01)
w
##   [1] 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14
##  [16] 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29
##  [31] 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44
##  [46] 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59
##  [61] 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74
##  [76] 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89
##  [91] 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00
# Se llenan las matrices por columna
x<-matrix(c(1,2,4,5),2,2)
x
##      [,1] [,2]
## [1,]    1    4
## [2,]    2    5
x<-matrix(c(1,2,4,5),2,2,byrow=TRUE)
x
##      [,1] [,2]
## [1,]    1    2
## [2,]    4    5
y<-matrix(c(2,2,4,4),2,2,byrow=TRUE)
y
##      [,1] [,2]
## [1,]    2    2
## [2,]    4    4
# Suma de matrices
x+y
##      [,1] [,2]
## [1,]    3    4
## [2,]    8    9
# MultilicaciĆ³n de matrices
x
##      [,1] [,2]
## [1,]    1    2
## [2,]    4    5
y
##      [,1] [,2]
## [1,]    2    2
## [2,]    4    4
# Producto de Hadamark que es componente a componente
x*y
##      [,1] [,2]
## [1,]    2    4
## [2,]   16   20
# Producto usual
x
##      [,1] [,2]
## [1,]    1    2
## [2,]    4    5
y
##      [,1] [,2]
## [1,]    2    2
## [2,]    4    4
x%*%y
##      [,1] [,2]
## [1,]   10   10
## [2,]   28   28
# r: random,d: density,p: probability,q: quantile
# X dis n(5,4)

x<-rnorm(1000,5,4)
x
##    [1]  3.474684607  7.420324679  1.497065391  7.250591268  5.505430831
##    [6] -0.869904754  4.924761998  1.310670333  3.534421789  4.060125175
##   [11]  6.399336838  2.780856408  6.921641748  9.246184146  6.320648930
##   [16]  5.492882088  8.407311475 -0.153929662  2.305490180  9.897853485
##   [21]  3.369758891  0.343401147  0.660497015 12.373540926 11.825486167
##   [26]  0.862447193 12.917949385  5.021511795 -3.066501955  8.176970392
##   [31]  2.484454984  5.525914344  5.107595334  5.672018536  3.977086886
##   [36]  6.081721848  4.389369623 11.815967387 12.036008244  9.612018170
##   [41]  5.657510763 11.122579926 -3.923438133  8.334116261  1.509181621
##   [46]  9.954526892  8.267050516 -6.052568321  5.311514173  6.653746860
##   [51] -6.481099056  4.157102117  2.840117878  7.665616795  0.754807329
##   [56]  8.778683892  7.987423323  5.740055775  2.438016618  2.558481112
##   [61]  0.815066238  0.009869510  8.580916228  2.726622912 14.122032220
##   [66]  5.746572507  9.998924120  5.071396091  2.723953276 12.678309922
##   [71] 11.429862139  8.415198506  5.564892518  5.022329627  4.273147057
##   [76] 10.625917593  0.494069412  9.389025885  2.760596292  2.501797009
##   [81]  3.519148228  8.129694755 10.548147308  8.289587345  6.782978913
##   [86]  4.561714535  4.101181437  7.012882611  5.242034491 14.994730170
##   [91] 11.156509604 10.139818161  6.028077721  4.332565966  4.125505267
##   [96]  3.670164889  4.155391182  3.382312213 13.880870290  1.503887789
##  [101]  6.746575764  6.211597363 -1.655514382  6.323724641  0.519895984
##  [106]  5.603494852  2.439794240 -2.112740676  1.979799669  3.059792769
##  [111] -3.116250478  3.912580021  0.800780424  2.923449179 14.529240559
##  [116]  6.664848263 11.770772376  0.391956734  8.154956534  3.947421029
##  [121] 10.166112097  9.372115789 14.928545180 14.573015969 10.840510801
##  [126]  8.447358151 10.094736282  1.805432560  6.193240088  6.433440756
##  [131]  7.098424263  4.451237197  3.662369169  6.728357107  5.210854343
##  [136]  2.481048032  2.356505158  3.758334162  3.914710910  8.615638113
##  [141]  9.208379645  3.357286796  0.841186496  7.657544923 11.618247988
##  [146]  6.773689764  3.805805984  1.782478310  5.375400588 -1.186153638
##  [151]  0.100002431 10.912913938  2.317429655  2.428805703  8.195846139
##  [156] 11.712462576  2.601481491  1.219431285  9.074008847  8.070501410
##  [161]  5.271895307 -5.143247176 -1.243744539  0.501552022  5.072428145
##  [166]  6.645936883 -1.459088700  6.504035815  6.024886219  0.419939338
##  [171]  2.206398487  3.783880929  7.254838476  9.038004605  2.953503632
##  [176]  0.001508494  2.734152478  0.154880983 10.216859831  0.964735545
##  [181]  2.833954423  6.822452207  2.119496524  6.127677372  7.487115259
##  [186]  7.256948114  4.149542198  0.592511203  8.020448253  8.883434985
##  [191]  7.692855199  4.352450962  5.432819648  7.999474763  1.776813767
##  [196] -0.480799556  2.583620500  9.137150632  0.523807937  6.808107200
##  [201] 10.513268308 -0.956290143  1.765208093  1.661385210  1.969397921
##  [206]  3.517201454  3.959808981  3.137083624  8.156857763  6.716057159
##  [211] 10.195654426  6.074401668 10.433946895  5.785500211  3.759708364
##  [216]  8.110674269  1.934440320  2.622930417  4.887601744  7.146307316
##  [221] 11.741147430 -3.695832824  8.982758986  9.065500018 10.266216548
##  [226]  3.953031038  7.431990568  3.736921509  6.735031467  5.084436785
##  [231]  5.734487881 11.475987310  1.662853252  7.585317256 -0.892456188
##  [236]  5.686093841  1.530705966  1.823912288 13.204580026  8.362904834
##  [241]  7.567341434  4.004384064  2.396380614 10.055334931  7.438607506
##  [246]  8.525369130  4.094050278  6.921369291  2.484918041 13.899449923
##  [251]  3.667600203  4.553185525  3.356375038 -5.898681305 14.201556458
##  [256]  7.400146448  6.851324837  4.218654150  3.774300302  7.126327559
##  [261]  4.343926208  5.099927019  5.217943137  5.387280880  7.102158530
##  [266]  1.932642329  1.160512752 14.231837573 -0.712154876  5.838095502
##  [271]  7.323898326  9.151806298  4.599542682  4.906072846 -0.164947968
##  [276]  3.578035692  4.961401317  6.520516760  5.634072443 -0.847030541
##  [281]  5.517459371  5.846357056  4.998742519  2.297417431  5.314570632
##  [286]  9.062258884 -0.752688349  6.660508497 -2.420185782  4.235866571
##  [291]  4.222010759  7.395041331  9.159081676  0.443597555  7.022868252
##  [296]  1.983800643  9.218371712  3.145094678  1.967727165  8.269209402
##  [301] -2.843536057  8.427618378  8.286263736  4.653670954  7.238362781
##  [306]  8.610306036  2.823853957  3.369508223 -2.408756486  4.359174314
##  [311]  5.463209264  5.586474926  4.627487195 -5.960382347  4.802349550
##  [316]  2.814545695  2.957929437  5.762580321  0.704319023  5.045323576
##  [321]  5.820351401  3.097356623 10.009627811  4.618118082  6.282734853
##  [326]  0.731450129  2.884001578 -0.618229631  4.193095838  9.147077707
##  [331]  7.905173145  8.863480148  6.330937562  9.561563611  1.966041699
##  [336] -0.166057251  9.807896757  3.512440391  3.332957700  1.917076192
##  [341]  5.127833553 11.587346492  4.626799294  2.219799911  8.010459753
##  [346]  9.858631856 10.989459696 12.812834229  7.130296645  4.725396695
##  [351]  8.957885254 11.305414235  7.433889717  8.713556344  5.475605768
##  [356]  8.840853256 -0.444654285  5.494318713  2.732750312  7.902191816
##  [361]  7.022360196  1.024888216  5.810934768  2.867132140  3.764079576
##  [366] -0.727749861  6.673871341  0.088575841  5.606422946 17.287677867
##  [371]  8.836772068  3.892282593  3.839883739  5.220662960  7.728351168
##  [376] 10.243688578  0.706239368 15.712084394  5.504886537  4.209484494
##  [381]  6.537559028 -0.133673468  1.960174984 11.640157389 10.647025002
##  [386] 11.382353271  4.951187261  7.523171542  5.227986195  0.381118959
##  [391]  3.489998998 12.443042371  2.388884999 -2.410416191  0.675521486
##  [396]  7.239130137  0.805632524  1.725984577  3.192737039  4.675745425
##  [401] -2.345148321 -0.690784724  2.685188096  2.320429152  0.758375117
##  [406]  1.843513851  8.307963656  8.720040807 -3.009462134 -0.479116753
##  [411] 10.083487003  5.910129404  5.230821204  8.967874907  8.700506165
##  [416]  0.842917972 -2.606668195  4.915078412  6.160260106  6.774399089
##  [421]  5.726156365  2.011996460  9.124478468  7.496799679  5.005363542
##  [426] 10.314044480  5.680551876  9.239251714  4.617354732  5.353141212
##  [431]  4.451604138  7.584221442  3.452390288  6.896654912  0.618945432
##  [436]  6.885587443  7.231196651  7.233470961  8.234094528  4.873752460
##  [441] -5.098529051  6.683488010  2.979570788  4.906770939 10.024115712
##  [446]  7.842031478  5.903590302  6.649856959  0.997047956  4.122194991
##  [451]  2.422129630  5.367626581  1.164070882  2.008527500  6.542069677
##  [456]  8.917622002  0.696295141  3.818390970  7.553193306  7.433751703
##  [461]  9.428435973  8.543780273 10.256464945  4.793992375  6.764957074
##  [466] -0.370977632 -4.329309678  7.755119629  8.743555890  2.945628899
##  [471]  9.568276241  6.216154959 11.587614226  4.303784757  3.909473208
##  [476] -4.420603149  3.980534497 -2.245261586  3.945892889 11.555989115
##  [481]  8.717515708  6.443818083  3.607840970  6.840440308  8.018215450
##  [486] -0.180141264  5.444214956  7.565679267  4.672794726 15.406444427
##  [491]  4.497075211  4.858157769 -1.678381000  5.507229087  5.658567020
##  [496]  7.430478304  6.841558794  5.115615555  4.176652688  9.603054704
##  [501]  8.020854917  8.186285321 12.798698314  6.918310528  4.210247758
##  [506]  2.403915852  8.230447854 -2.709790634  1.674512893  2.950784325
##  [511] 11.816994186  3.405982521 11.192403062 -0.146870876  6.217149989
##  [516]  6.090694239  5.170615338  2.976716266  2.951355052 10.031035046
##  [521]  9.919007382  5.759660705  4.734321004  7.883428130  5.913672183
##  [526]  6.082320738  5.769711466  2.989829830  8.806197497  9.078666174
##  [531]  5.745042527  3.132755470  7.762807368 10.027936894  3.192759945
##  [536] -5.916990112  2.185752370  0.529030879  2.955331724  7.602555535
##  [541]  4.138300611  5.521052060  6.688745908  1.901278875  2.773768613
##  [546]  3.829644701  4.206834108 -1.593613197  4.618540545  3.113942712
##  [551] 10.818196078  0.974173814 -1.014562586  4.954760637  3.899933994
##  [556]  6.868549495  4.618265759  2.184050490  4.708785107  8.404759348
##  [561]  3.940141051 10.433709200  5.154754131  8.512160204  2.214268008
##  [566]  4.261659349  1.653100252  5.702980357 12.938623928  6.619547371
##  [571]  5.134526236 10.081113473  4.404191685  4.653959889  6.048944999
##  [576]  8.057723006  6.576099846  8.324826856  3.655384150  3.081625458
##  [581]  9.562542680 -0.319619698 11.465537580  1.085959886  0.755594105
##  [586]  3.559216799  8.354000094 -0.119672060  9.770242985  6.535790192
##  [591] -7.389972735  1.829239193 -5.389231363  7.049675973  5.891100084
##  [596] 13.274963980  3.373535016  6.798734333  3.298325596 10.326967093
##  [601] -0.590279924  5.186184543  7.845412786  3.439168935  6.067870067
##  [606]  7.671929309  7.602586450  5.372539695  2.728278353  4.576858540
##  [611]  1.398358347  0.928762157  6.452381610  6.654496583  3.716555107
##  [616]  3.532775366  5.527044125  6.218741656 11.432415684  2.055109229
##  [621]  5.296212428  8.367318543  6.434536169  7.698399145 -1.650668988
##  [626]  3.454684759 -2.743837255  2.614079536  8.591819824  3.687501227
##  [631]  8.667224128  2.684197112  4.710921085  2.731757027  2.150516700
##  [636]  2.890599102  3.715345845  2.759072854  5.430537395  5.098936081
##  [641]  9.019628887  3.601499633  8.307239729  3.244257724  4.373001570
##  [646]  3.018793801 10.090721271  3.783136902  3.681011113  5.649504234
##  [651]  7.147510826  6.973192631  1.263714240  1.681162538  8.110163495
##  [656]  8.234624392  4.388560075 -3.261852662  1.738727482  6.814841655
##  [661]  6.954317219 -3.613015017  3.542129806 -0.072089604 -3.162180873
##  [666]  4.113368100  0.712525929  2.436815496 13.263809387 -0.871371750
##  [671] 12.245804489 13.518846376 -0.851818147  8.216186887  8.449397767
##  [676]  5.415257269 17.349612477  0.356392152 15.472960307  7.101077891
##  [681]  3.815936951  3.094980546  2.439994315  7.478276393  3.096243892
##  [686]  5.053231304 14.879353582  1.802682751  3.071746741  1.868082927
##  [691]  5.885017396  4.283000487  7.135403159  5.651580662  2.101150304
##  [696] 12.872907457  1.215896888  5.110016304  0.203470043  9.156608868
##  [701] -2.221126394  5.471211202  9.116589747  0.323290685  8.184850296
##  [706]  5.761433845  8.860027809  5.183217738  7.805600709  3.009759619
##  [711]  2.722219292  0.565026727  9.005868927 10.101233075  5.489099563
##  [716]  9.848628853  3.395089177 -1.575366984  4.121825934  5.060681014
##  [721]  4.771359234  4.214638632  3.554705748  6.118828778  5.486861591
##  [726]  7.790611910  7.882193408  5.959535638 -5.012127092  1.256878234
##  [731]  0.970614757  5.008125872  1.766548313  1.645074028  2.952096383
##  [736] -0.655718092 12.951219662  4.065900216  2.452670233 10.953595859
##  [741]  9.641336493  2.894473185  3.226197547  1.155357462  0.649989788
##  [746]  8.811084400  3.334824277  0.356117715 -1.138516998  4.394900969
##  [751]  3.343288841  4.852498956  9.550363093  2.291658959  9.129457655
##  [756] -1.317702658  6.765891552  9.650197451 -1.101282567 10.991773769
##  [761]  7.826754875  5.087042826  1.050662963  5.822671947  6.292012195
##  [766] -2.518951028  4.164256940 -1.224053305  6.955994967  2.298789237
##  [771]  5.718643755  7.243494729  8.859188619  0.122431024  9.883996294
##  [776]  0.886674497  4.889454779  4.999487915  3.948810785  2.409690830
##  [781]  1.057953272  5.495498262  8.017224668  3.861031547 11.240687909
##  [786]  6.100356514  2.368753976  0.279802212  5.051307674  7.041432532
##  [791]  2.166001745  4.822420064  3.840579714  4.113078733 13.449768232
##  [796]  2.934041032 13.162428628  1.645426888  5.230886876 -0.871205289
##  [801]  4.778108489 -3.026376338  6.217275528  3.157579704  8.682008520
##  [806]  4.411757225 -2.039229657  0.391802938  0.449919380 10.300903650
##  [811]  1.819532991  5.286792530 11.335357528 14.089813053 -4.845476727
##  [816]  2.557085189  8.459297342  5.939535804  7.527137072 11.242477040
##  [821]  0.989890205  9.369594645  7.345734223  8.047307080 17.059172908
##  [826]  9.878837811  7.201253419  5.811901736  7.207769182  8.239191552
##  [831]  1.074925242  7.294188953  9.006603748  1.024722074  3.220611389
##  [836] 10.119492601 -2.439713666  4.655454984  5.920093383  1.563714952
##  [841]  4.275179944  6.904698174  4.311741639  4.652600674  2.582039465
##  [846]  0.723752404  7.203417334  4.494263313  8.634397198  9.279822425
##  [851]  5.160577525  4.878713448  5.230026710  1.149569578  3.695891295
##  [856]  5.231218643  4.938796563  8.238646735  3.265597426 -1.853584163
##  [861]  9.282827364  5.343791691  4.680209020  9.976851211  9.494380188
##  [866]  3.818348649  0.154355261  4.004300465  2.446360872  3.019868205
##  [871]  5.109214796  7.723345661  7.471977426 -1.183607753  6.782769000
##  [876]  1.655877716  8.041444431  6.387424058  8.499541823 14.675856942
##  [881] 10.395210700  4.447490988  7.158008412  8.835359061  1.853395880
##  [886]  9.031709482 -0.606756821  2.617722886 10.144704905  2.161739876
##  [891]  8.270177352  5.479660524  1.563030202 13.289218757  1.110683475
##  [896]  2.563566120  1.882011534  2.933380333  2.850559799  2.396213929
##  [901]  4.303748870 13.690721601 -0.683054729  4.738278764 19.408022827
##  [906]  3.055395868  4.404999692  3.706874427  4.021972911  5.438293355
##  [911]  8.063105208  1.950343969  9.388070884  2.144806265  6.168456173
##  [916] 12.661866274  1.725200406  3.330222521  4.184312649  8.573376908
##  [921]  4.039502950  4.953202425 13.120163138  2.470905637  7.690100833
##  [926]  1.981812962  3.693149332  2.367644287  2.165955266  4.932925980
##  [931]  8.098090376  2.965308970  5.392025881  7.106269657 -3.692152502
##  [936] -5.243902923  3.291836904  6.565840272  5.413951554  4.158325457
##  [941] 12.069404622  7.858483347  3.490915179  7.836663572  6.211309242
##  [946]  4.330783729  6.565789619 -3.989350722  6.448832082  4.766521464
##  [951]  6.667190664  5.958198133  6.658178292 12.160930460  0.943289927
##  [956]  3.443517330  9.718226820 -2.901116862  3.749262028 10.587377072
##  [961]  8.812941225 -3.396794109  3.108451398 -2.820347413  8.162148021
##  [966]  8.906330122  1.816033888  5.529857506  7.453444380  1.075977052
##  [971]  1.677373125 13.219803036  1.550584865  7.118097171  1.825535790
##  [976]  1.865518616  2.215736078 10.488214397  4.265683081 12.172528617
##  [981]  7.104590277  6.063179835  5.434980037  6.019365398 -1.162694594
##  [986]  3.825358549  9.087823473  0.346732202 11.901322130  3.746388918
##  [991]  8.194163827  5.664653301  6.273865689  5.859097223  2.428277331
##  [996] 10.823179666 10.146051669 -6.143899844  2.103211628 15.679884218
hist(x,col="green", main = "Normal", xlab = "x", ylab = "f(x)")

mean(x)
## [1] 5.112178
var(x)
## [1] 16.07808
sd(x)
## [1] 4.009748
sqrt(var(x))
## [1] 4.009748
# Booleano
sd(x) == sqrt(var(x))
## [1] TRUE
plot(density(x),col = 4,main = "Normal", xlab = "x", ylab = "f(x)")

#RegresiĆ³n

#MĆ­nimos cuadrados
# Modelo con ruido
x<-rnorm(1000)
y<-3+4*x+rnorm(1000)
plot(x,y,col = "pink",main = "MĆ­nimos cuadrados", xlab = "x", ylab = "y")

# Modelo sin ruido
x<-rnorm(1000)
y<-3+4*x
plot(x,y,col = "pink",main = "MĆ­nimos cuadrados", xlab = "x", ylab = "y")

# Ajustar el modelo

# FunciĆ³n lm que significa linear model
x<-rnorm(1000)
y<-3+4*x+rnorm(1000)
plot(x,y,col = "pink",main = "MĆ­nimos cuadrados", xlab = "x", ylab = "y")
fit <- lm(y~x)
fit
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##       2.997        4.005
abline(a=fit$coefficients[1],b=fit$coefficients[2],col="red")
# Linea de tendencia
abline(a=fit$coefficients[1],b=fit$coefficients[2],col="red", lwd=4)
abline(a=fit$coefficients[1],b=fit$coefficients[2],col="blue",lwd=2)
abline(h=mean(y), col="green",lwd=3)
abline(v=mean(x), col="green",lwd=3)

# HipĆ³tesis nula es el supuesto
# Se rechaza la hipĆ³tesis nula cuando el p valor es menor a la significancia
# 0.05,0.1,0.01
summary(fit)
## 
## Call:
## lm(formula = y ~ x)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.5040 -0.6733  0.0087  0.7009  2.9315 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  2.99747    0.03105   96.55   <2e-16 ***
## x            4.00522    0.03181  125.89   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9817 on 998 degrees of freedom
## Multiple R-squared:  0.9408, Adjusted R-squared:  0.9407 
## F-statistic: 1.585e+04 on 1 and 998 DF,  p-value: < 2.2e-16
# Cuando no es significativa la pendiente
x<-rnorm(100)
y<-rnorm(100)
plot(x,y,col = "pink",main = "MĆ­nimos cuadrados", xlab = "x", ylab = "y")
fit <- lm(y~x)
fit
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##     0.06002      0.11476
abline(a=fit$coefficients[1],b=fit$coefficients[2],col="red")
# Linea de tendencia
abline(a=fit$coefficients[1],b=fit$coefficients[2],col="red", lwd=4)
abline(a=fit$coefficients[1],b=fit$coefficients[2],col="blue",lwd=2)
abline(h=mean(y), col="green",lwd=3)
abline(v=mean(x), col="green",lwd=3)

summary(fit)
## 
## Call:
## lm(formula = y ~ x)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.19186 -0.57793 -0.04307  0.75648  1.83191 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)  0.06002    0.09159   0.655    0.514
## x            0.11476    0.09417   1.219    0.226
## 
## Residual standard error: 0.9159 on 98 degrees of freedom
## Multiple R-squared:  0.01493,    Adjusted R-squared:  0.004878 
## F-statistic: 1.485 on 1 and 98 DF,  p-value: 0.2259
# Muestras
n<-100
M<-1000
# 1000 MUESTRAS DE TAMANO 50
x<-matrix(0,M,n)
for(i in 1:M){
  x[i,]<-rnorm(n,10,5)
}
x[3,]
##   [1]  3.9547369  2.7875097  2.1830356 15.6413358  7.5002046  6.0780192
##   [7]  8.4639051 15.4223452 16.0870471  9.8378319 -1.9946417 11.5757415
##  [13] 12.9842590  4.8046507 21.3222847 11.4266954 14.1729545 10.9165296
##  [19]  4.9992239  8.7133368  8.4376911  4.0426209  9.7037017  7.7464041
##  [25] 12.2799406 10.9881136 10.8803684 18.0400112 10.0007561  1.8813501
##  [31]  8.9013298  4.5244028 17.3161877 10.9757577 18.7486787 11.4219377
##  [37] 10.9008949 14.4175634  8.5205152  8.7210158 16.1161525  6.8679379
##  [43] 15.4017994 14.6376113 14.4220641  9.6195275 12.0377898 12.5670214
##  [49]  7.1416386  4.8125833  1.4749882 17.3714727  4.3004531 14.4412115
##  [55] 15.2746374  7.8205159 11.4009882 18.9217117 -5.5693181 -2.8597923
##  [61]  3.6090296  7.7841812 10.1047158  6.9464918  7.1623729 20.2520082
##  [67]  8.0596571  4.1900791 12.4405334  5.5716484 10.6971190  9.9043427
##  [73] 18.3577226  6.9665901  8.9145065  8.8728488  9.6254917 10.8248334
##  [79] 11.1455817  9.1658912  6.6468923  9.8737295 15.6345347  0.4627328
##  [85]  9.1913951 13.9857265  0.9009201  3.9876103 12.0842630 11.5273570
##  [91] 12.8901213 13.3229064 12.2397129  9.6374052 14.6790291  1.5060640
##  [97]  7.7931549 11.3262602  6.1182429 16.5083841
media <- NULL
for(i in 1:M){
  media[i]<-mean(x[i,])
}
5/sqrt(100)
## [1] 0.5
sd(media)
## [1] 0.4911002
# Ley de los grandes nĆŗmeros con la Gamma

x<-list()
for(i in 1:1000){
  x[[i]]<-rgamma(i,4)
}
media <- 4
x[[40]]
##  [1] 3.3241414 4.8549465 3.2620400 9.1495405 6.7140924 6.1835605 3.5663004
##  [8] 1.2892236 7.2314978 7.5830485 2.0570160 5.1286404 5.2307057 2.5999791
## [15] 7.8384555 3.8205046 4.1050864 4.6130689 5.1438422 2.4811335 2.2901102
## [22] 6.3324445 5.3681996 6.3564208 1.1738929 3.7395107 7.1112280 7.3480222
## [29] 3.1954881 2.9860373 2.4475868 0.9959772 3.3571860 5.2694622 6.0395473
## [36] 8.5358811 5.2555421 5.6356117 4.4953366 5.4866053
med<-NULL
for(i in 1:1000){
  med[i]<-mean(x[[i]])
}
plot(med,type = "l", main = "Ley de los grandes nĆŗmeros", col = "green")
abline(h=4,col ="red")

x<-list()
for(i in 1:10000){
  x[[i]]<-rgamma(i,4)
}
media <- 4
x[[40]]
##  [1] 4.4311637 0.9811048 1.3617644 1.8264549 6.7915183 1.4258221 5.0335751
##  [8] 5.7618521 5.0464608 6.1702201 2.6983879 2.7529414 4.2730520 2.0103390
## [15] 6.9440839 1.9727104 4.0929725 6.1933153 2.2557611 4.9158585 3.4038416
## [22] 5.2139902 5.0012427 3.2899398 4.4521857 3.5161474 4.9602348 4.5923395
## [29] 1.5342736 1.7384536 4.6023518 4.1206112 4.2402248 2.5245972 2.6043424
## [36] 4.5966140 3.5571406 4.1669671 2.9074034 2.4074323
med<-NULL
for(i in 1:10000){
  med[i]<-mean(x[[i]])
}
plot(med,type = "l", main = "Ley de los grandes nĆŗmeros", col = "green")
abline(h=4,col ="red")

x<-list()
for(i in 1:10000){
  x[[i]]<-rgamma(i,4)
}
media <- 4
x[[40]]
##  [1]  2.1067737  2.8001803  3.3536619  8.0004113  1.3262565  2.5954911
##  [7]  6.4855941  5.5069171  2.1043177  4.9054853  3.2731280  6.0263047
## [13]  0.9630069  1.9073788  2.6548557  2.6736820  3.1470006  1.1005844
## [19]  6.0722955  7.7407331  2.4557978  6.5747815  3.2426123  3.6893152
## [25]  3.1935128  1.7240427  2.7627988 10.0432123  4.7276073  2.6964527
## [31]  1.8387272  8.2956991  4.8459083  6.5991379  7.2047243  2.4513162
## [37]  2.8550662  1.9260916  3.2204721  1.4080509
med<-NULL
for(i in 1:10000){
  med[i]<-mean(x[[i]])
}
plot(med,type = "l", main = "Ley de los grandes nĆŗmeros", col = "green")
abline(h=4,col ="red")

# Teorema de lĆ­mite central

# Gamma

M<-1000
n<-100
data<-matrix(0,M,n)
for(i in 1:M){
  data[i,]<-rgamma(n,4)
}
med<-NULL
for(i in 1:M){
  med[i]<-mean(data[i,])
}
hist(med, col = "green")

# Cauchy

M<-1000
n<-100
data<-matrix(0,M,n)
for(i in 1:M){
  data[i,]<-rcauchy(n,4)
}
med<-NULL
for(i in 1:M){
  med[i]<-mean(data[i,])
}
hist(med, col = "green")

density(med)
## 
## Call:
##  density.default(x = med)
## 
## Data: med (1000 obs.);   Bandwidth 'bw' = 0.3373
## 
##        x                y           
##  Min.   :-187.1   Min.   :0.000000  
##  1st Qu.: 744.8   1st Qu.:0.000000  
##  Median :1676.7   Median :0.000000  
##  Mean   :1676.7   Mean   :0.002311  
##  3rd Qu.:2608.6   3rd Qu.:0.000000  
##  Max.   :3540.5   Max.   :0.625499
plot(density(med))

# No tiene momentos pero no contradice el TLC

# POIS

M<-1000
n<-100
data<-matrix(0,M,n)
for(i in 1:M){
  data[i,]<-rpois(n,4)
}
med<-NULL
for(i in 1:M){
  med[i]<-mean(data[i,])
}
hist(med, col = "green")

density(med)
## 
## Call:
##  density.default(x = med)
## 
## Data: med (1000 obs.);   Bandwidth 'bw' = 0.04552
## 
##        x               y            
##  Min.   :3.203   Min.   :0.0001096  
##  1st Qu.:3.594   1st Qu.:0.0394735  
##  Median :3.985   Median :0.3251355  
##  Mean   :3.985   Mean   :0.6384972  
##  3rd Qu.:4.376   3rd Qu.:1.1988249  
##  Max.   :4.767   Max.   :1.9357555
plot(density(med))