# hw-7

Homework 7 in STAT400: Computational Statistics @ CSU

## Assignment

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

1. 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?

2. 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.

3. 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 2 ($$\phi_1$$ 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 rstudio.cloud will also be used in grading, so be sure they are identical and reproducible.