hw-6

Homework 6 in STAT400: Computational Statistics @ CSU

Assignment

Be sure to set.seed(400) at the beginning of your homework.

  1. Develop two Monte Carlo integration approaches to estimate \(\int\limits_0^5 x^2 \exp(-x) dx\). (You must use different distributions in the two approaches). Check your answer using the integrate() function.

  2. Estimating the cdf of a normal distribution. Use \(m = 1000\).

    1. Implement all 3 methods that we discussed in class (Example 1.7, Page 9-10 of Ch. 6 Notes) to estimate the cdf of a normal distribution \(\Phi(x)\). Note that you will need to show some derivations for method 2.
    2. Compare your estimates with the output from the R function pnorm() for \(x = 0.5, 1, 2, 3\). Summarise your findings comparing the performance of the methods.
    3. For each method, compute an estimate of the variance of your Monte Carlo estimate of \(\Phi(2)\). Summarise your findings.
    4. For each method, compute a \(95\%\) confidence interval for \(\Phi(2)\). Summarise your findings. Which CI is narrower and why does that matter?

  3. Compute a Monte Carlo estimate \(\hat{\theta}_1\) of \[ \theta = \int\limits_0^{0.5} e^{-x} dx \] by sampling from the Uniform(\(0, 0.5\)) and estimate the variance of \(\hat{\theta}_1\). Find another Monte Carlo estimator \(\hat{\theta}_2\) by sampling from the Exponential(1) distribution and estimating its variance.

    Which of the variances (of \(\hat{\theta}_1\) or \(\hat{\theta}_2\)) is smaller?

Turn in in a pdf of your homework to canvas. Your .Rmd file on the server will also be used in grading, so be sure they are identical and reproducible.