hw-7

Homework 7 in STAT400: Computational Statistics @ CSU

Assignment

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

1. Find two importance functions \(\phi_1\) and \(\phi_2\) that are supported on \((1, \infty)\) and are “close” to \[ h(x) = \frac{x^2}{\sqrt{2\pi}} e^{-x^2/2}, \qquad x > 1. \]

   Which of your two importance functions should produce the smallest variance in estimating \[ \int\limits_1^\infty \frac{x^2}{\sqrt{2\pi}} e^{-x^2/2} dx \] by importance sampling? Explain.

   Hint: You will need to create plots of \(\phi_1\), \(\phi_2\), and \(g(x)f(x)\) as well as \(g(x)f(x)/\phi(x)\) to answer this question.

2. Obtain a Monte Carlo estimate of
   \[ \int\limits_1^\infty \frac{x^2}{\sqrt{2\pi}} e^{-x^2/2} dx \] using importance sampling with the two importance sampling functions you chose in Problem 4 (\(\phi_1\) and \(\phi_2\)). Obtain an estimate of the variance for each and compare.

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.

Be sure to share your server project with the instructor and grader:

1. Open your hw-7 project on liberator.stat.colostate.edu

2. Click the drop down on the project (top right side) > Share Project…

   

3. Click the drop down and add “stat400instructors” to your project.

   

This is how you receive points for reproducibility on your homework!