#Investigate the properties of linear combination
# of Random Variables

#Suppose X1 is Normally distributed N(10,1)
#and X2 is Normally distributed N(20,5)

Investigate the properties of X+X2 and X-X2

X1 = rnorm(500, mean=10, sd = 1)
X2 = rnorm(500, mean=20, sd = 5)
Y1 = X1 + X2 

Y2=X1-X2

par(mfrow=c(2,2))
hist(X1,probability=T,xlim=range(Y1,Y2), main="X1 ~ Normal(10,1)")
#title(main="X1 ~ Normal(10,1)")

hist(X2,probability=T, xlim=range(Y1,Y2), main="X2 ~ Normal(20,5)")
#title(main="X2 ~ Normal(20,5)")

hist(Y1,probability=T, main="Y = X1+X2", xlim=range(Y1,Y2))
#title(main="Y = X1+X2")

hist(Y2,probability=T, xlim=range(Y1,Y2), main="Y = X1-X2")
#title(main="Y = X1-X2")
###############################


X1 = rgamma(500, shape=1,scale=1)
X2 = rnorm(500, mean=20, sd = 5)
Y1 = X1 + X2 

Y2=X1-X2

par(mfrow=c(2,2))
hist(X1,probability=T,xlim=range(Y1,Y2), main="X1 ~ Normal(10,1)")
#title(main="X1 ~ Normal(10,1)")

hist(X2,probability=T, xlim=range(Y1,Y2), main="X2 ~ Normal(20,5)")
#title(main="X2 ~ Normal(20,5)")

hist(Y1,probability=T, main="Y = X1+X2", xlim=range(Y1,Y2))
#title(main="Y = X1+X2")

hist(Y2,probability=T, xlim=range(Y1,Y2), main="Y = X1-X2")
#title(main="Y = X1-X2")
#################################