#R version 3.3.2 
#Tema Dranca Constantin 334
#Exercitiul 1

#x + x^3 + x^5 = 3 u
#x(1+x^2+x^4) = 3u
#Fie F(x) = (F1(x) + F2(x) + F3(x))/3
#unde F1(x) =x ; F2(x) = x^3; F3(x) = x^5
#pentru F1, F2 si F3 vom avea repatitiile uniforme asociate U1, U2, U3 cu U = U1 + U2 +U3
#vom obtine astfel: F1(x) +F2(x) +F3(x) =3U1 +3U2 +3U3
# F1(x) = x = 3u1 
# F2(x) = x^3 = 3u2 => x = (3u2)^(1/3)
# F3(x) = x^5 = 3u3 => x = (3u3)^(1/5)
U1 <- runif(1000)
U2 <- runif(1000)
U3 <- runif(1000)

X1 <- 3*U1
X2 <- (3*U2)^(1/3)
X3 <- (3*U3)^(1/5)

X <- X1 + X2 + X3
hist(X)


#exercitiul 2:
#F(x) e continua in punctul x=1
# x -> inf => F(x) -> 1
# y = 1-(1/x) => x= -1/(y-1)
# x-> inf => y ->1
# x-> 0 => y ->0
# deci, x in [0,inf) => y in [0,1)
# F(-1/(y-1)) = {1-
#