# 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.