#################
### SESTI CAS ###
#################

#2.
uzorak1 <- rpois(n = 50, lambda = 3) # simulacija uzorka
uzorak1
x1 <- seq(0, 5, 0.2)
plot(x1, x1^sum(uzorak1)/prod(factorial(uzorak1))*exp(-x1*length(uzorak1)), main = "Funkcija verodostojnosti", xlab = "lambda", ylab = "L(lambda)", type = "l")
mean(uzorak1)

#3.
uzorak2 <- rexp(n = 50, rate = 2)
uzorak2
x2 <- seq(0, 5, 0.2)
plot(x2, x2^length(uzorak1)*exp(-x1*sum(uzorak2)), main = "Funkcija verodostojnosti", xlab = "lambda", ylab = "L(lambda)", type = "l")
1/mean(uzorak2)

#9. 
simulacija <- function(n, beta, k){
  brojac = 0
  podaci <- replicate(k, rnorm(20, 3, 1))
  for (i in 1:k) {
    xn = mean(podaci[1:20,i]) #x srednje
    sn2 = var(podaci[1:20,i]) #s^2
    c <- qt((1+beta)/2, 19)
    interval_poverenja <- c(xn -c*sqrt(sn2/n), xn +c*sqrt(sn2/n))
    if (findInterval(3, interval_poverenja) == 1){
      brojac = brojac + 1
    }
  }
  return(brojac)
}
simulacija(20, 0.93, 10000)/10000


#11.
uzorak <- c(rep(0,3), rep(1,6), rep(2,5), rep(3,2),4)
uzorak
length(uzorak)
mean(uzorak)
var(uzorak)
sd(uzorak)
mean(uzorak)/sd(uzorak)*sqrt(17) # realizovana vrednost test statistike
qt(0.95,16) 
2*(1-pt(5.61,16)) # p vrednost
t.test(uzorak, mu = 0, alternative = "two.sided", conf.level = 0.95)

#12.
uzorak <- c(rep(5,3), rep(10,8), rep(15,11), rep(20,5), rep(25,3))
(mean(uzorak)-14)/sd(uzorak)*sqrt(length(uzorak)) # realizovana vrednost test statistike
qt(0.95, df = 29) 
1-pt(0.487, 29) # p vrednost
t.test(uzorak, mu = 14, alternative = "greater", conf.level = 0.95)