# 7.1
x   <- c(102, 99, 106, 103, 96, 98, 100, 105, 103, 98, 104, 107)
(n  <- length(x))
(m  <- mean(x))
(s2 <- var(x))
(s <- sd(x))

# distribucni funkce
(x <- sort(x))
(t <- unique(x))
nt <- length(t)
 
cetnost <- NULL                   # prazdny vektor
for(i in 1:nt){                   # pro i od 1 do 10 vloz na i-tou pozici vektoru 'cetnost' pocet hodnot vektoru X, ktere jsou mensi nebo rovny i-te hodnote ve vektoru t
  cetnost[i] <- sum(x <= t[i])}

Fx <- cetnost/n                   # vyberova distribucni funkce

xx <- c(min(t)-1, t, max(t)+1)
yy <- c(0, Fx, 1)  

#graf distibucni funkce
plot(xx, yy, type='n', xlab='x', ylab='y', main='F(x)')   # prazdny graf
lines(xx, yy, type='s', col='red', lwd=2)                 # schodovita cara
arrows(96, 0, 95, 0, col='red', lwd=2, length=0.1)        # dolni sipka
arrows(107, 1, 108, 1, col='red', lwd=2, length=0.1)      # horni sipka


#7.2
x <- c(10, 16, 5, 10, 12, 8, 4, 6, 5, 4)
mean(x)
var(x)
sd(x)

sort(x)
sum(x>8.5)/length(x)  # P(X.8.5)

#7.3
x <- c(1, 4, 5, 9, 11, 13, 23, 23, 28)
y <- c(64, 71, 54, 81, 76, 93, 77, 95, 109)

cov(x,y) # vyberova kovariance
cor(x,y) # vyberovy korelacni koeficient

# 7.4 a)
# 99% oboustranny IS pro mu, kdyz sigma^2 zname
m     <- 3000
sigma <- 20
n     <- 16
alpha <- 0.01

(dh <- m-sigma/sqrt(n)*qnorm(1-alpha/2))
(hh <- m-sigma/sqrt(n)*qnorm(alpha/2))

# b) 90% levostranny IS pro mu kdyz sigma^2 zname
alpha <- 0.1
(dd <- m-sigma/sqrt(n)*qnorm(1-alpha))

# c) 95% pravostranny IS pro mu kdyz sigma^2 zname
alpha <- 0.05
(hh <- m-sigma/sqrt(n)*qnorm(alpha))