Verjetnostne funkcije

pnorm(1.96)
## [1] 0.9750021
pnorm(-3:3)
## [1] 0.001349898 0.022750132 0.158655254 0.500000000 0.841344746 0.977249868
## [7] 0.998650102
seq(-3,3,0.1)
##  [1] -3.0 -2.9 -2.8 -2.7 -2.6 -2.5 -2.4 -2.3 -2.2 -2.1 -2.0 -1.9 -1.8 -1.7
## [15] -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1.0 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3
## [29] -0.2 -0.1  0.0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1.0  1.1
## [43]  1.2  1.3  1.4  1.5  1.6  1.7  1.8  1.9  2.0  2.1  2.2  2.3  2.4  2.5
## [57]  2.6  2.7  2.8  2.9  3.0
x <- seq(-3,by=0.1,to=3)
plot(x,pnorm(x))

plot(x,dnorm(x))

q <- qnorm(c(0.1,0.5,0.9))
q
## [1] -1.281552  0.000000  1.281552
plot(x,dnorm(x))
abline(v=q,col="red")

Izbiranje slučajnih vrednosti

n <- 5
rnorm(n)
## [1] -0.1329935  1.1775975 -1.3663690 -0.1689643 -0.4314434
n <- 4
set.seed(234)
rnorm(n)
## [1]  0.6607697 -2.0529830 -1.4992061  1.4712331
rnorm(n)
## [1]  1.4591385  0.1401390  0.2091844 -3.0360898
set.seed(234)
rnorm(n)
## [1]  0.6607697 -2.0529830 -1.4992061  1.4712331

vzorčna porazdelitev povprečij

Parametri simulacije

n <- 25    # velikost vzorca
N <- 500  # število ponovitev vtorčenja

Osnova za eno vzorčenje

rnorm(n)
##  [1]  1.45913853  0.14013904  0.20918439 -3.03608982 -0.48693413
##  [6] -1.08786731  0.05785971  1.10397550 -0.02561697  0.51484639
## [11]  0.99005668  0.30345432 -0.93007223  0.08403068  0.52677963
## [16]  0.01586862  0.20535184  1.01628335  0.40899897 -0.70523547
## [21]  0.23561433  0.34513003 -1.01979955 -1.41182225 -1.36384225

Povprečje enega vzorca

mean(rnorm(n))
## [1] -0.02759944
xbars <- replicate(N,mean(rnorm(n)))
head(xbars)
## [1]  0.08604292  0.12522377 -0.33267657 -0.12583228 -0.01012376  0.04905540

Če tisto kar ponavljam da več kot eno vrednost.

replicate(10,range(rnorm(n)))
##           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]
## [1,] -2.787564 -2.241600 -2.522571 -1.954202 -2.019224 -2.378163 -1.304047
## [2,]  1.429279  2.667244  1.877631  1.183542  1.986979  1.948687  2.555177
##           [,8]      [,9]     [,10]
## [1,] -2.428939 -2.775687 -1.040065
## [2,]  2.438534  1.746891  1.473830

Malo risanja

plot(xbars)
abline(h=0,col="red")

plot(xbars,ylim=c(-3,3))
abline(h=0,col="red")

Standardna napaka ocene je standardni odklon vzorčne porazdelitve (ocen).

sd(xbars)
## [1] 0.2068568

Funkcija za simulacijo

simula <- function(n=9,N=200){
  xbars <- replicate(N,mean(rnorm(n)))
  plot(xbars,ylim=c(-3,3),main=n)
  abline(h=0,col="red")
  text(10,2,round(sd(xbars),3),adj=0)
}

Razne simulacije

simula()

simula(25)

simula(4,1000)

Za več velikosti vzorcev

ns <- c(1,2,4,16,25,100,400)
for(n in ns) simula(n)