#1)
library(MASS)

#exponencialni model
lambda<-fitdistr(data, "exponential")$estimate[1]

#normalni model
mu<-fitdistr(data, "normal")$estimate[1]
sigma<-fitdistr(data, "normal")$estimate[2]

#lognormalni model
muL<-fitdistr(data, "lognormal")$estimate[1]
sigmaL<-fitdistr(data, "lognormal")$estimate[2]

#logisticky model
loc<-fitdistr(data, "logistic")$estimate[1]
scale<-fitdistr(data, "logistic")$estimate[2]

#a)
x1<-100000

1-ecdf(data)(x1)
1-pexp(x1,lambda)
1-pnorm(x1,mu,sigma)
1-plnorm(x1,muL,sigmaL)
1-plogis(x1,loc,scale)


#b)
x2<-1000000

1-ecdf(data)(x2)
1-pexp(x2,lambda)
1-pnorm(x2,mu,sigma)
1-plnorm(x2,muL,sigmaL)
1-plogis(x2,loc,scale)

#c)
q1<-0.999

quantile(data,q1)
qexp(q1,lambda)
qnorm(q1,mu,sigma)
qlnorm(q1,muL,sigmaL)
qlogis(q1,loc,scale)

#d)
q2<-0.9999

quantile(data,q2)
qexp(q2,lambda)
qnorm(q2,mu,sigma)
qlnorm(q2,muL,sigmaL)
qlogis(q2,loc,scale)

#2.)
data<-c(rep(0,4),rep(1,6),rep(2,6),rep(3,8),rep(4,3),6,6,10)
n<-length(data)
table(data)/n

lambda<-fitdistr(data,"Poisson")$estimate[1]
lambda

#a) 
x<-(0:10)

#graf relativnich cetnosti a odhadnute pravd. funkce
plot(table(data)/n, type='p',ylim=c(0,0.3),ylab='pravdepodobnost')
points(dpois(x,lambda)~x,col=2,type='h')

#graf empiricke a odhadnute distribucni funkce
plot(ecdf(data),main="Empiricka distribucni funkce")
curve(ppois(x,lambda),col=2,add=TRUE)

#graf rozdilu obou funkci
D<-function(x){
D<-ecdf(data)(x)- ppois(x,lambda)
}

curve(D,0,15,col=2,ylim=c(-0.07,0.07),ylab="D(x)")
abline(h=0)

#b)
#spocitame ocekavane cetnosti pro hodnoty 0-10
dpois(x,lambda)*n

#musime nektere hodnoty sloucit: 0,1,2,3,4,5+
k<-6               #pocet trid
prob<-c(dpois(c(0,1,2,3,4),lambda),1-ppois(4,lambda)) #teoreticke pravdepodobnosti
hk<-c(4,6,6,8,3,3) #pozorovane cetnosti

#porovnani ocekavanych a pozorovanych cetnosti
n*prob
hk

chisq.test(hk,p=prob) #test pouzijeme jen pro vypocet testove statistiky (nesedi stupne volnosti))
X2<-chisq.test(hk,p=prob)$statistic
1-pchisq(X2,k-2)  #hledana p-hodnota