######################
##  Harjoitukset 3  ##
######################

#########
## 3.1 ##
#########

odotus<-scan("Odotusaika.txt")
m<-mean(odotus)   # 7.93311
v<-var(odotus)    # 14.25054
a<-m^2/v          # 4.416269
b<-v/m            # 1.796338
x<-seq(0,25,0.5)
hist(odotus,freq=F)
lines(x,dgamma(x,shape=a,scale=b))
plot(density(odotus))
curve(dgamma(x,shape=a,scale=b),0,25,add=T,lty=2)


#########
## 3.4 ##
#########

## (a)
uf<-function(l){-(n*(l-y)^2)/2}
y<-3
n<-20
curve(uf(x),1,5,ylim=c(-40,0))
n<-40
curve(uf(x),1,5,add=T,lty=2)
legend(5, 0, c("n=20","n=40"),lty=c(1, 2), xjust=1, yjust=1)


#########
## 3.7 ##
#########

## (a)
uf1<-function(l){-n*l+n*y*log(l)}
uf2<-function(l){n*y*(log(y)-1)-(n*(l-y)^2)/(2*y)}
n<-10
y<-exp(1)
curve(uf1(x),0,6,ylim=c(-15,0))
curve(uf2(x),0,6,add=T,lty=2)
legend(6, 0, c("tarkka","likiarvo"),lty=c(1, 2), xjust=1, yjust=1)

## (b)
n<-10
y<-25
max1<-uf1(y)   #554.719
max2<-uf2(y)   #554.719
uf1u<-function(l){-n*l+n*y*log(l)-max1}
uf2u<-function(l){n*y*(log(y)-1)-(n*(l-y)^2)/(2*y)-max2}
curve(uf1u(x),20,30)
curve(uf2u(x),20,30,add=T,lty=2)
legend(30, 0, c("tarkka","likiarvo"),lty=c(1, 2), xjust=1, yjust=1)