# Blatt 9

#########
## Aufgabe 9.1
## a)
print(abs(pi^exp(1) - exp(pi)), digits=12)
## b)
n <- c(10,100,1000)
b <- sqrt(n+sqrt(n)) - sqrt(n-sqrt(n))
print(b)
## c)
n <- 20
p <- 0.3
k <- 0:n
pvec <- choose(n,k)*p^k*(1-p)^(n-k)
print(pvec)
pvec2 <- dbinom(k, n, p)
print(pvec2)

#########
## Aufgabe 9.2
N <- 1000
r <- runif(N)
n <- 1:N
mr <- cumsum(r)/n
plot(n,mr)

#########
## Aufgabe 9.3
## a)
N <- 500
x <- runif(N,-1,1)
y <- runif(N,-1,1)
dev.new() # neues Grafikfenster
plot(x,y)
## b)
sel <-  x^2 + y^2 + x*y < 1 
u <- x[sel]
v <- y[sel]
dev.new()
plot(u,v)
## c)
rxy <- sum(x*y)/(sqrt(sum(x^2))*sqrt(sum(y^2)))
ruv <- sum(u*v)/(sqrt(sum(u^2))*sqrt(sum(v^2)))
print(rxy)
print(ruv)