## R-Code fuer Blatt 3


## Aufgabe 3


# 1.Graphik
# Simulation der Daten für Aufgabe auf dem Blatt und Graphik
par(mfrow=c(1,1))
set.seed(20100506)
x_1 <- rpois(10000,1) # 10000 Zufallszahlen ziehen aus Po(1), d.h. Gewicht 4/5
x_2 <- rpois(2500,5)  # 2500 Zufallszahlen ziehen aus Po(5), d.h. Gewicht 1/5
z <- c(x_1,x_2)
hist(z, breaks=0:(max(z)+1)-0.5)
mean(z); var(z)

# 2. Graphik
set.seed(20100506)
# Simulation der Daten für Aufgabe auf dem Blatt und Graphik
x_1 <- rpois(10000,1) # 10000 Zufallszahlen ziehen aus Po(1), d.h. Gewicht 4/5
x_2 <- rpois(2500,5)  # 2500 Zufallszahlen ziehen aus Po(5), d.h. Gewicht 1/5
z <- c(x_1,x_2)
par(mfrow=c(3,1))
hist(z, breaks=0:(max(z)+1)-0.5, ylim=c(0,4000))
hist(x_1, breaks=0:(max(z)+1)-0.5, ylim=c(0,4000))
hist(x_2, breaks=0:(max(z)+1)-0.5, ylim=c(0,4000))
mean(z); var(z)



## Aufgabe 4 a), b), c)

## 1000 Realisationen von X ~ N(0,1)

par(mfrow=c(1,1))

set.seed(667) 

X<-rnorm(1000)

## 500 Realisationen von Y ~ N(10,3)

set.seed(665) 
 
Y<-rnorm(500,10,3)

## Zusammenfassen

Z<-c(X,Y)

## Histogramm

hist(Z,freq=F,xlim=c(min(Z),max(Z)),ylim=c(0,.3),breaks=25)

## empirische Momente

m<-mean(Z)
s<-sd(z)

## einfache normalverteilung basierend auf empirischen Momenten

curve(dnorm(x, mean = m, sd = s), from = -5, to = 20, n = 1000, type = "l", col="green", lwd=2, lty=1,add=T)

## gemischte Normalverteilungen mit Mischverhältnis entsprechend der Simulation

curve((2/3)*dnorm(x, mean = 0, sd = 1)+(1/3)*dnorm(x, mean = 10, sd = 3), from = -5, to = 20, n = 1000, add = TRUE, type = "l", col="red", lwd=2)


## Aufgabe 4e)

mu <- -2
sigma <- 1

par(mfcol=c(3,2))
curve(dnorm(x, mean = 0, sd = 1), from = -5, to = 15, n = 1000, type = "l", ylab = "density", col="green", lwd=2, lty=2, main=paste("Mischung mit mu=", eval(mu), " und sigma=", eval(sigma)))
curve(dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="blue", lwd=2, lty=2)
curve(0.5*dnorm(x, mean = 0, sd = 1)+0.5*dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="red", lwd=2)

mu <- 5
sigma <- 1

curve(dnorm(x, mean = 0, sd = 1), from = -5, to = 15, n = 1000, type = "l", ylab = "density", col="green", lwd=2, lty=2, main=paste("Mischung mit mu=", eval(mu), " und sigma=", eval(sigma)))
curve(dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="blue", lwd=2, lty=2)
curve(0.5*dnorm(x, mean = 0, sd = 1)+0.5*dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="red", lwd=2)

mu <- 10
sigma <- 1

curve(dnorm(x, mean = 0, sd = 1), from = -5, to = 15, n = 1000, type = "l", ylab = "density", col="green", lwd=2, lty=2, main=paste("Mischung mit mu=", eval(mu), " und sigma=", eval(sigma)))
curve(dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="blue", lwd=2, lty=2)
curve(0.5*dnorm(x, mean = 0, sd = 1)+0.5*dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="red", lwd=2)

mu <- -2
sigma <- 2

curve(dnorm(x, mean = 0, sd = 1), from = -5, to = 15, n = 1000, type = "l", ylab = "density", col="green", lwd=2, lty=2, main=paste("Mischung mit mu=", eval(mu), " und sigma=", eval(sigma)))
curve(dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="blue", lwd=2, lty=2)
curve(0.5*dnorm(x, mean = 0, sd = 1)+0.5*dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="red", lwd=2)

mu <- 5
sigma <- 2

curve(dnorm(x, mean = 0, sd = 1), from = -5, to = 15, n = 1000, type = "l", ylab = "density", col="green", lwd=2, lty=2, main=paste("Mischung mit mu=", eval(mu), " und sigma=", eval(sigma)))
curve(dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="blue", lwd=2, lty=2)
curve(0.5*dnorm(x, mean = 0, sd = 1)+0.5*dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="red", lwd=2)

mu <- 5
sigma <- 3

curve(dnorm(x, mean = 0, sd = 1), from = -5, to = 15, n = 1000, type = "l", ylab = "density", col="green", lwd=2, lty=2, main=paste("Mischung mit mu=", eval(mu), " und sigma=", eval(sigma)))
curve(dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="blue", lwd=2, lty=2)
curve(0.5*dnorm(x, mean = 0, sd = 1)+0.5*dnorm(x, mean = mu, sd = sigma), from = -5, to = 15, n = 1000, add = TRUE, type = "l", col="red", lwd=2)