# Jakaumia

Binomijakauma
n<- 40; p<- 1/4

x<-29:31
sum(dbinom(x,n,p))
pbinom(31,n,p)-pbinom(28,n,p)  # toinen tapa

plot(dbinom(x,n,p))

plot(x,dbinom(x,n,p))

===================
# Geometrinen jakauma
# R:ssä lasketaan ennen 1. onnistumista sattuvien epäonnistumisten lukumäärää x: 0,1,...
# Yritysten määrä on x+1

x<-1:15; p<-0.4
plot(x,dgeom(x-1,p),type="h")

par(mfrow=c(2,2))
p<-0.2; plot(x,dgeom(x-1,p),type="h",ylim=c(0,0.8))
p<-0.4; plot(x,dgeom(x-1,p),type="h",ylim=c(0,0.8))
p<-0.6; plot(x,dgeom(x-1,p),type="h",ylim=c(0,0.8))
p<-0.8; plot(x,dgeom(x-1,p),type="h",ylim=c(0,0.8))

# Olkoon Y yritysten määrä
# P(Y<= 5), kun p<-0.04
# P(Y<= 5)=P(Y-1=<= 5-1)=P(X=<= 4)

p<-0.4
pgeom(4,p)

p<-0.2
# P(2<=Y<= 5)=P(1<=x<= 4)
x<-1:4
sum(dgeom(x,p))

# P(2<=Y<= 5)=P(1<=x<= 4)=P(0<x<= 4)
pgeom(4,p)-pgeom(0,p)

# Negatiivinen binomijakauma
############################


dnbinom(x, r, p)

r<-4; p<-0.6
x<-4:7
sum(dnbinom(x-4, r, p))

p<-0.5
sum(dnbinom(x-4, r, p))

plot(x,dnbinom(x-4, r, p),type="h")

p<-0.6
plot(x,dnbinom(x-4, r, p),type="h")

########################
# Binomi-Poissonin likiarvo
a)
n<-25; k<-25
x<-0:4
dbinom(x,n,1/k)
#[1] 0.36039672 0.37541325 0.18770662 0.05996184 0.01374125
# E(X)=n/k=1

dpois(x,1)
#[1] 0.36787944 0.36787944 0.18393972 0.06131324 0.01532831
dbinom(x,25,1/25)/dpois(x,1)
# [1] 0.9796598 1.0204790 1.0204790 0.9779590 0.8964625

(b)
n<-50; k<-50
x<-0:4
dbinom(x,n,1/k)
#[1] 0.36416968 0.37160171 0.18580086 0.06066967 0.01454834

dpois(x,1)
#[1] 0.36787944 0.36787944 0.18393972 0.06131324 0.01532831

dbinom(x,n,1/k)/dpois(x,1)
#[1] 0.9899158 1.0101182 1.0101182 0.9895035 0.9491156

###############################
# Hypergeometrinen jakauma

dhyper(x, a, b, n)
# a valkoisten pallojen lkm, b mustien pallojen lkm 
# n on otoskoko
a<-3; b<-7; n<-5
x<-0:3		# valkoisten pallojen lukumäärät otoksessa
dhyper(x, a, b, n)

x<-0:5
dhyper(x, a, b, n)