#zuweisen, abfragen, rechnen

3+4
2*pi
x<-4
x
x<-c(1,3,5,2)
x
x<-3:7
x<-seq(1,3,0.2)
x
x<-seq(1,7,length = 21)
x

#einfache Funktionen

mean(x)
sum(x)
var(x)
sd(x)

#Kombinatorik

factorial(5)
factorial(100)

choose(4,2)
choose(12,3)


#Verteilungen diskret

dhyper(4,10,12,6) #vier Erfolge, bei 10 Treffern, 12 Nieten und 6 Versuchen
dbinom(3,13,0.17) 
pbinom(3,13,0.17)
sum(dbinom(0:3,13,0.17))
pbinom(7,13, 0.17) - pbinom(2,13,0.17)
sum(dbinom(3:7,13,0.17))
dpois(14,20)
ppois(14,20)
ppois(30,20)

#Approximation grafisch

plot(pbinom(0:20,20,0.33), type = "l", ylab ="Wahrscheinlichkeit")
lines(phyper(0:20,7,14,20), col = "blue")
lines(phyper(0:20,10,20,20), col = "green")
lines(phyper(0:20,20,40,20), col = "yellow")
lines(phyper(0:20,50,100,20), col = "red")

#plot
plot(ppois(0:24,4), type = "l", ylab ="Wahrscheinlichkeit")
lines(pbinom(0:24,24,1/6), col = "blue")


#Verteilungen stetig

pnorm(0)
dnorm(0)
pnorm(0,1,4)
pnorm(2)
pnorm(2) - pnorm(-2)

punif(4,2,12)
dunif(seq(0,1,0.1))

plot(dnorm, xlim = c(-4,4))
plot(pnorm, xlim = c(-4,4))

plot(dexp, xlim = c(0,5))
lines(seq(0,5,0.01),dexp(seq(0,5,0.01), rate = 0.5), type = "l", col = "green")
lines(seq(0,5,0.01),dexp(seq(0,5,0.01), rate = 5), type = "l", col = "red")

plot(pexp, xlim = c(0,5))
lines(seq(0,5,0.01),pexp(seq(0,5,0.01), rate = 0.5), type = "l", col = "green")
lines(seq(0,5,0.01),pexp(seq(0,5,0.01), rate = 5), type = "l", col = "red")

# Konzepte

x<-runif(100,0,5)
mean(x)
var(x)
((5-0)^2)/12
sd(x)

y<-rnorm(100,2,3)
plot(x,y, pch  =19, xlab = "Gleichverteilung", ylab  ="Normalverteilung")
cov(x,y)
cor(x,y)

z<-x+rnorm(100,0,1)
plot(x,z, pch  =19, xlab = "Gleichverteilung", ylab  ="Gleichverteilung mit Rauschen")
cov(x,z)
cor(x,z)


#Grafiken

library(reshape)
data(tips)
attach(tips)
head(tips)

#Barchart
barplot(table(size), col = "grey")

#Streudiagram
plot(total_bill, tip, pch = 19, xlab = "Gesamtrechnung", ylab = "Trinkgeld")
points(total_bill[total_bill<10], tip[total_bill < 10], col = "magenta", pch =19)
points(total_bill[total_bill<10 & tip > 5], tip[total_bill < 10 & tip > 5], col = "green", pch =19)

#Histogram
hist(tip, col = "grey")
hist(tip, col = "grey", breaks = 15)


#Ordnungsaufgabe
x<-vector(length = 10000)
y<-vector(length = 10000)

for(i in 1:10000){
v<-runif(5)
v<-sort(v)
x[i]<-v[5]
y[i]<-v[4]
}
mean(x)
mean(y)

for(i in 1:10000){
v<-runif(20)
v<-sort(v)
x[i]<-v[20]
y[i]<-v[19]
}
mean(x)
mean(y)

for(i in 1:10000){
v<-runif(100)
v<-sort(v)
x[i]<-v[100]
y[i]<-v[99]
}
mean(x)
mean(y)