## Simulate Data
reps = 1e5
y.vec = cos(runif(n=reps, min=-pi/2, max=pi/2))
hist(y.vec)

## plot f(x) = cos(x)
x=seq(from=-pi/2, to=pi/2, by=0.01)
fx = cos(x)
plot(x, fx, type="l")


est.mean = mean(y.vec)
est.mean

est.var = var(y.vec)
est.var



## plot f(x) = cos(x) and g(x) = 1-0.5 x^2
x=seq(from=-pi/2, to=pi/2, by=0.01)
fx = cos(x)
gx = 1-0.5*x^2
hx = array(1, length(x) )
plot(x, fx, type="l", lty = 1)
lines(x, gx, lty=2)
lines(x, hx, lty=3)
legend(x=1, y=1, legend=c("f(x)", "g(x)", "h(x)"), lty=c(1,2,3))


reps=1e5
A=0
E = var(cos(runif(n=reps, min=-pi/2, max=pi/2)))
rel.err = abs((A-E)/E)

reps=1e5
A=cos(0)
E = mean(cos(runif(n=reps, min=-pi/2, max=pi/2)))
c(A,E)

reps=1e5
A=1-pi^2/24
E = mean(cos(runif(n=reps, min=-pi/2, max=pi/2)))
c(A,E)






## Variance Approximation/Exact with relative error
reps=1e5
A= (1/12)*(1/5.5)^2
E = var(log(runif(n=reps, min=5, max=6)))
rel.err = abs((A-E)/E)
c(A,E)
rel.err

## Crude Expected Value Approximation
reps=1e5
A=log(5.5)
E = mean(log(runif(n=reps, min=5, max=6)))
c(A,E)


## Second order Expected Value Approximation
reps=1e5
A=log(5.5) + 0.5*(1/12)*(-1/(5.5)^2)
E = mean(log(runif(n=reps, min=5, max=6)))
c(A,E)