dir()

#=====================================================
#                  P R I K L A D   1
#=====================================================

n <- 0:12
m.obs <- c(3, 24, 104, 286, 670, 1033, 1343, 1112, 829, 478, 181, 45, 7)
M <- sum(m.obs)
p <- sum((n * m.obs)) / (12 * 6115)
(p <- round(p, 4)) # pst narozeni chlapce v rodine s 12 detmi je 0.5192 (51.92%).

#=====================================================
#                  P R I K L A D   2
#=====================================================

m.ocek <- dbinom(0:12, 12, 0.5192) * 6115
(m.ocek <- round(m.ocek))
tab <- data.frame(rbind(m.obs, m.ocek))
names(tab) <- 0:12 
tab # tabulka pozorovanych cetnosti (prvni radek) a ocekavanych cetnosti (druhy radek) za predpokladu, ze X ~ Bin(12, 0.5192)

# Graf porovnavajici pozorovane a ocekavane pocetnosti
plot(0:12, m.obs, type = 'h', col = 'red', las = 1, 
     xlab = 'pocet chlapcu', ylab = 'abs. cetnost')
points(0:12, m.obs, col = 'red', pch = 20)      # pozorovane pocetnosti (namerena data)
lines(0:12 + 0.1, m.ocek, type = 'h', lty = 2)  # ocekavane pocetnosti za predpokladu, ze X ~ Bin(12, 0.5192)
points(0:12+0.1, m.ocek, pch = 20)
legend('topright', pch = c(20, 20), col = c('red', 'black'),
       legend = c('pozorovane', 'ocekavane'), bty = 'n')

#=====================================================
#                  P R I K L A D   3
#=====================================================

# (a)
dbinom(9, 12, 0.5192) # Pr(X = 9) ... pravdepodobnosti funkce

# (b)
pbinom(4, 12, 0.5192) # Pr(X <= 4) ... distribucni funkce

# (c)
1-pbinom(7, 12, 0.5192) # Pr(X >= 8) = 1 - Pr(X < 8) = 1 - Pr(X <= 7) ... 1 - distribucni funkce

# (d)
sum(dbinom(4:7, 12, 0.5192)) # Pr(4 <= X <= 7) = Pr(X = 4) + Pr(X = 5) + Pr(X = 6) + Pr(X = 7)
pbinom(7, 12, 0.5192) - pbinom(3, 12, 0.5192) # analogicky vypocet: # Pr(4 <= X <= 7) = Pr(X <= 7) - Pr(X <= 3)


#=====================================================
#                  P R I K L A D   4
#=====================================================

# graf pravdepodobnosti funkce
px <- dbinom(0:12, 12, 0.5192)
plot(0:12, px, type = 'h', ylab = 'p(x)', las = 1, xlab = '')
mtext('x', side = 1, line = 2.1)
mtext('N = 12, p =0.5192', side = 1, line = 3.2)
points(0:12, px, col = 'red', pch = 19)


# graf distribucni funkce
Fx <- pbinom(0:12, 12, 0.5192)
plot(0:12, Fx, xlab = '', ylab = 'F(x)', xlim = c(-1, 13),
     ylim = c(0, 1), las = 1, type = 'n')
segments(0:12, Fx, 1:13, Fx)
arrows(0,0, -1, 0, length= 0.1)
arrows(12, 1, 13, 1, length = 0.1)
points(0:12, c(0, Fx[1:12]), pch = 21, col = 'brown', bg = 'white')
points(0:12, Fx, col = 'red', pch = 20)
mtext('x', side = 1, line = 2.1)
mtext('N = 12, p =0.5192', side = 1, line = 3.2)