##########
# Blatt2 #
##########

#aufgabe 2

data(michelson)
attach(michelson)

#a
boxplot(Speed ~ Expt)

#b
exp_1<-Speed[Expt ==1]

summary(exp_1)

IQR(exp_1)

max(exp_1) - min(exp_1)

sd(exp_1)
sqrt(1/(length(exp_1)-1) * sum((exp_1 - mean(exp_1))^2)) 

#c
dotplot(exp_1) #library(lattice) 
hist(exp_1, col = c(rep("grey",8),"magenta"), breaks = 8)
boxplot(exp_1)
points(min(exp_1), col = "magenta", pch =20)

#Ausreisser nicht klar definierbar:
#faerben in dotplot
plot(exp_1, rep(0, 20), pch = 19)
points(c(min(exp_1), max(exp_1)), c(0,0), col = "magenta", pch = 20) 

#aufgabe 3

#d
n<-456
k<-110
#Transformation auf ein Zehntel der Originalwerte, sonst numerische Instabilitaet

p<-seq(0.01, 0.99, 0.01)


L<- p^k * (1-p)^(n-k)

plot(p,L,type = "l")
abline(v = k/n)

#aufgabe 5

x_1<-rpois(100,2)
x_2<-rpois(1000,2)
x_3<-rpois(10000,2)

(est_1<-mean(x_1))
(est_2<-mean(x_2))
(est_3<-mean(x_3))


(bias_1<-(est_1 - 2))
(bias_2<-(est_2 - 2))
(bias_3<-(est_3 - 2))

(var_1<-est_1 / 100)
(var_2<-est_2 / 1000)
(var_3<-est_3 / 10000)

(mse_1<-bias_1^2 + var_1)
(mse_2<-bias_2^2 + var_2)
(mse_3<-bias_3^2 + var_3)


kons_1<-rep(0,100)
for(i in 1:100){
kons_1[i]<-mean(rpois(100,2))	
}

kons_2<-rep(0,100)
for(i in 1:100){
kons_2[i]<-mean(rpois(1000,2))	
}

kons_3<-rep(0,100)
for(i in 1:100){
kons_3[i]<-mean(rpois(10000,2))	
}

par(mfrow = c(3,1))
hist(kons_1, xlim = c(1.3,2.7))
hist(kons_2, xlim = c(1.3,2.7))
hist(kons_3, xlim = c(1.3,2.7))