5.1


The exact value is:

\[ \theta = \int_{0}^{\frac{\pi}{3}} sin(x) \; dx\] \[ -cos(x) |_{0}^{\frac{\pi}{3}}\] \[ cos(0)-cos(\frac{\pi}{3}) \]


The estimate is \[ \hat{\theta} = \overline{g_{n}(X)} = \frac{1}{n}\sum_{x = 1}^{n} g(X_{i})\] \[ = (b-a)\overline{g_{n}(X)}\] \[ = \frac{\pi}{3} (\overline{sin({x_{i}}})\]

set.seed(12345)
n <- 10000
x <- runif(n, 0, pi/3)
theta.hat <- (pi/3) * mean(sin(x))
exact <- -(cos(pi/3)-cos(0))

kable(cbind(c("exact","estimate"),round(c(theta.hat,exact),3))) %>%
   kable_paper("hover", full_width = F)
exact 0.501
estimate 0.5


5.2


\[\Phi(x) = \int_{-\infty}^{x} \frac{1}{2\pi}e^{\frac{-t^{2}}{2}} \; dt \] We need to break it into two cases, \[x \ge 0 , x < 0 \] Substituting y=t/x, we have \[ \theta = \int_{0}^{1}xe^{-(xy)^{2}/2}dy \] The estimate is \[ \hat{\theta} = \overline{g_{m}(u)} = \frac{1}{m}\sum_{x = 1}^{m} xe^{-(xu_i)^{2}/2}\]

set.seed(12345)
x <- seq(0.5,2.5, length = 10)
m <- 10000
u <- runif(m)
cdf <- numeric(length(x))
for (i in 1:length(x)) {
  g <- x[i] * exp(-u* x[i]^2/2)
  cdf[i] <- mean(g) / sqrt(2*pi) + 0.5
}

Phi <- pnorm(x)
kable(round(rbind(x, cdf, Phi),3)) %>%
   kable_paper("hover", full_width = F)
x 0.500 0.722 0.944 1.167 1.389 1.611 1.833 2.056 2.278 2.500
cdf 0.687 0.754 0.804 0.837 0.855 0.859 0.853 0.840 0.823 0.804
Phi 0.691 0.765 0.828 0.878 0.918 0.946 0.967 0.980 0.989 0.994 
samples <- 2*exp(-2^2*u^2/2)/sqrt(2*pi)+.5
estimate <- mean(samples)
v <- var(samples)/m
LB <- estimate-1.96*sqrt(v)
UB <- estimate+1.96*sqrt(v)
kable(cbind(c("Estimate","Variance estimate","Lower Bound","Upper Bound"),c(round(estimate,3),round(v,8),round(LB,3),round(UB,3)))) %>%
  kable_paper("hover",full_width = F)
Estimate 0.977
Variance estimate 5.25e-06
Lower Bound 0.972
Upper Bound 0.981