# Luento 30.9.04
# Esimerkki 4.2 (Tehtävä 5.4)
# Valinta palauttamatta. Valitaan arpa n kertaa. Hatussa N arpaa
#######################
N<-20; n<-3
r<-4
fx<- function(r) {(r/N)^n-((r-1)/N)^4}
Fx<- function(r) {(r/N)^n}
r<-1:20; y<-fx(r)
plot(r,y,type="h")

y<-cumsum(y); y<-Fx(r)
plot(r,y,type="s")

#####################
# Valinta palauttaen
#####################

Fy<-function(r) {choose(r,n)/choose(N,n)}
fy<-function(r) {Fy(r)-Fy(r-1)}
r<-1:20; y<-fy(r)
plot(r,y,type="h")

y<-Fx(r)
plot(r,y,type="s")

####################
# Binomijakauma (4.2)
n<-7; p<-0.15
x<-1
dbinom(x,n,p) #0.396007
x<-3
pbinom(x,n,p)  # 0.9878968
# pbinom(q, size, prob, lower.tail = TRUE, log.p = FALSE)
1-pbinom(1,n,p) # P(X>=2)= 1-P(X<=)= 0.2834159
pbinom(1, n, p, lower.tail = F)