######################
##  Harjoitukset 1  ##
######################


#########
## 1.5 ##
#########

## (a)

curve(dnorm(x,8.78,sqrt(0.16)),7.5,10,xlim=c(7.5,10),
ylim=c(0,3.1),ylab="density",lty=1, lwd=2)

curve(dnorm(x,8.78,sqrt(0.16/9)),7.5,10,add=TRUE,lty=2,lwd=2)

legend(10, 3.1, c("X",expression(bar(X))),lty=c(1, 2),
xjust=1,yjust=1,lwd=2)


## (b)
curve(dchisq(x,8),0,25,ylab="density",lty=1,lwd=2)

qchisq(0.95,8)
b<-(qchisq(0.95,8)*0.16)/8
qchisq(0.05,8)
a<-(qchisq(0.05,8)*0.16)/8


#########
## 1.7 ##
#########

b<-(1-exp(-5/3))^4*(5*exp(-5/3)+1-exp(-5/3))
c<-exp(-5/3)


#########
## 1.8 ##
#########

## (a)
a<-qchisq(0.95,15)

## (b)
pnorm(((a/15)-1)/(sqrt(2/15)))
pnorm((a-15)/(sqrt(30)))


#########
## 1.9 ##
#########

curve(dchisq(x,15),0,35,xlim=c(0,35),ylim=c(0,0.08),
ylab="density",lty=1,lwd=2)

curve(dnorm(x,15,sqrt(30)),0,35,add=TRUE,lty=2,lwd=2)

legend(35, 0.08,c("Y","approximaatio"),lty=c(1, 2),
xjust=1,yjust=1,lwd=2)