##b)
#Suppose $X_1$ and $X_3$ are independent standard normal random variables.
n = 100
x1<-rnorm(n,0,1)
x3<-rnorm(n,0,1)

#find the correlated x2 as follows:
rho<-0.5
x2<-rho*x1+sqrt(1-rho^2)*x3
r = cor(x1,x2)  #the empirical pearson correlation


## c)
y<-x1+x2+rnorm(n,0,1)
mod1<-lm(y~x1)
summary(mod1)

#finding the information directly from R:
est_b1 = summary(mod1)$coef[2,1]
est_sd_b1 = summary(mod1)$coef[2,2]

#calculating the information by your self:
X           = model.matrix(mod1)
est_b1      = (solve(t(X)%*%X)%*%t(X)%*%y)[2]
est_sigma   = sqrt(crossprod(mod1$res)/(n-2))
est_sd_b1   = est_sigma*sqrt(solve(t(X)%*%X)[2,2]) 	
	#= est_sigma*sqrt( 1/(var(x1)*(n-1))  )
###taking the t-test:
tobs1<-(est_b1-1)/est_sd_b1
pval1<-2*(1-pt(abs(tobs1),98));
tobs2<-(est_b1-1.5)/est_sd_b1
pval2<-2*(1-pt(abs(tobs2),98));

pval1   #reject H0: b_1 != 1.
pval2   #accepting H0: b_1 = 1.5


## d)
mod_true<-lm(y~x1+x2)
summary(mod_true)

s1<-sqrt(var(x1))
s2<-sqrt(var(x2))

mod_true$coeff[2]+r*s2/s1*mod_true$coeff[3]
est_b1
#the same value