Normalverteilung

Die Temperatur in einem Kühlhaus zur Lagerung von subtropischen Früchten sei normalverteilt mit Erwartungswert \(\mu\) = 7 und Varianz \(\sigma^2\) = 4, d.h. sie wird geeignet durch den Wahrscheinlichkeitsraum \[(\mathbb{R}, \mathcal{B}(\mathbb{R}), \mathcal{N}_{μ, \sigma^2}).\]Was ist die Wahrscheinlichkeit, dass die Temperatur im akzeptablen Bereich zwischen 5 °C und 13 °C liegt?

#sigma <- sqrt(n*p*(1-p))
#mu <- n*p
sigma.squared <- 4
sigma <- sqrt(sigma.squared)
mu <- 7
 
from <- 5
to <- 13
 
const <- 1/sqrt(2*pi*sigma^2)
integrand <- function(x) {const * exp(-(((x-mu)^2)/(2*sigma^2)))}
 
solution <- integrate(integrand, from, to)
solution

## 0.8399948 with absolute error < 2.7e-12

Plot

library(plotly)


 #Draw plot
#Transform plot data into data frame
range <- 2*sigma
df <- data.frame(seq(from-2*range, to+range, .1),integrand(seq(from-2*range, to+range, .1)))
colnames(df)<- c('x','y')

#Area Under the Curve
#First subst the data and add the coordinates to make it shade to y = 0
auc <- subset(df, x >from & x < to)

p1 <- ggplot(data=df, aes(x=x, y=y)) + 
  geom_line() + 
  geom_area(data=auc, fill="#099dd9", alpha=.5) +
  scale_x_continuous(breaks=seq(from-3,to+2,2))+
  xlab("Temperatur in Grad Celsius") +
  ylab("Wahrscheinlichkeitsdichte") + 
  ggtitle("Wahrscheinlichkeitsdichtefunktion:\n Normalverteilung")

#Uncomment to see the ggplot2 plot
#p1

# Creating a fancy plotly plot
plot_ly(p1)

Normierung

Verteilungsfunktion der Standardnormalverteilung

mu <- 7
sigma.squared <- 4
sigma <- sqrt(sigma.squared)
a <- 5 
b <- 13
to.lower <- (b-mu)/sigma
to.upper <- (a-mu)/sigma
from <- -Inf
c <- 1/sqrt(2*pi)
integrand <- function(t) {c * exp(-.5*(t^2))}
 
integrate(integrand, from, to.lower)$value - integrate(integrand,from, to.upper)$value

## [1] 0.8399948

#Draw plot
#Transform plot data into data frame
range <- sigma
df <- data.frame(seq(to.upper-2*range, to.lower+range, .1),integrand(seq(to.upper-2*range, to.lower+range, .1)))
colnames(df)<- c('x','y')


#Area Under the Curve
#First subst the data and add the coordinates to make it shade to y = 0
auc <- subset(df, x >to.upper & x < to.lower)

p2 <- ggplot(data=df, aes(x=x, y=y)) + 
  geom_line() + 
  geom_area(data=auc, fill="#099dd9", alpha=.5) +
  scale_x_continuous(breaks=seq(to.upper-2,to.lower,1))+
  xlab("z-Wert") +
  ylab("Phi(x)") + 
  ggtitle("Wahrscheinlichkeitsdichtefunktion:\n Standardnormalverteilung")

#Uncomment to see the ggplot2 plot
#p2

# Creating a fancy plotly plot
plot_ly(p2)

Abschätzung durch Binomial- und Poissonverteilung

Wie wahrscheinlich ist es, dass unter n = 100 Wuerfen mit einem Wuerfel zwischen 12 und 22 Sechser auftreten?

Binomialverteilung

n <-100
p <- 1/6
k<- 12:22
binom <-  0
for(i in k)
  binom = binom + choose(n,i) * p^i * (1-p)^(n-i)
 
binom

## [1] 0.8592305

Stirlingsapproximation for big n factorials

stirling <- function (n){
  sqrt(2*pi*n) * (n/exp(1))^n
}

Poissonverteilung

n <- 100
p <- 1/6
lambda <- n*p
k<- 12:22
 
poisson <- 0
for(i in k)
  poisson <- poisson + exp(-lambda) * (lambda^i)/stirling(i) 
 
poisson

## [1] 0.8254448

Normalverteilung

n = 100
p = 1/6
 
sigma <- sqrt(n*p*(1-p))
mu <- n*p

from <- 12
to <- 22
range <- sigma 

const <- 1/sqrt(2*pi*sigma^2)
integrand <- function(x) {const * exp(-(((x-mu)^2)/(2*sigma^2)))}
 
solution <- integrate(integrand, from, to)
solution

## 0.818548 with absolute error < 9.1e-15

Plot

#Transform plot data into data frame
df <- data.frame(seq(from-2*range, to+1.5*range, .1),integrand(seq(from-2*range, to+1.5*range, .1)))
colnames(df)<- c('x','y')

#Area Under the Curve
#First subst the data and add the coordinates to make it shade to y = 0
auc <- subset(df, x >from & x < to)

p3 <- ggplot(data=df, aes(x=x, y=y)) + 
  geom_line() + 
  geom_area(data=auc, fill="#099dd9", alpha=.5) +
  scale_x_continuous(breaks=seq(from-3, to+1,2))+
  xlab("Anzahl Treffer (Augenzahl 6)") +
  ylab("Wahrscheinlichkeitsdichte") + 
  ggtitle("Wahrscheinlichkeitsdichtefunktion - Normalverteilung")

#Uncomment to see the ggplot2 plot
#p3

# Creating a fancy plotly plot
plot_ly(p3)

plotly_POST(p3, filename = "r-docs/normal_distribution")

## No encoding supplied: defaulting to UTF-8.

## Success! Modified your plotly here -> https://plot.ly/~slackoverflow/19

plotly_POST(p2, filename = "r-docs/standard_normal_distribution")

## No encoding supplied: defaulting to UTF-8.

## Success! Modified your plotly here -> https://plot.ly/~slackoverflow/17