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\centerline {\bf Monte Carlo Methods - Spring 16 - Homework 3 } 

\bigskip

\no 1. (antithetic) Consider the network example. 

\smallskip
\no (a) In class and in the notes I used the antithetic random variable 
$Y=f(1-U)$ where $1-U$ means $(1-U_1,1-U_2,1-U_3,1-U_4,1-U_5)$.
I find that the variance is reduce from about $0.1577$ for MC without
the antithetic variable to $0.01823$ with the antithetic RV. 
Reproduce this result. 

\smallskip
\no (b) Now let $\tilde{U}=(1-U_1,U_2,U_3,1-U_4,U_5)$ and let 
$Y=f(\tilde U)$. Note that $\tilde{U}$ has the same joint distribution 
as $U$, so $Y$ is still a valid antithetic RV. 
How much variance reduction do you get? Can you explain this?
What happens if I do some other tranformation of the probability space, e.g., 
\\
$\tilde{U}=(U_1,1-U_2,1-U_3,U_4,U_5)$ ?

\bigskip

\no 2. (control variates) In the network example in class I took
$Y=T_1+T_4$ and then just set $\alpha=1$ so that the RV we do MC on 
is just $Z=X-(Y-1)$. I find that the variance is reduced from about 
$0.1577$ for MC without a control variate to 
$0.0412$ with this control variate. 
Now let $Z=X-\alpha(Y-1)$. 
What is the optimal $\alpha$ and how much variance reduction do you get?

\bigskip

\no 3. (control variates) From Fishman's {\it First course...}
We want to compute the following integral by Monte Carlo:
\bea
\int_0^1 e^{-x \cos(\pi x)} \, dx = E[f(U)]
\eea
where $U$ is uniform on $[0,1]$ and $f(x)=e^{-x \cos(\pi x)}$.
Suppose we use the control variate $Y=g(U)$ with $g(x)=e^{-x}$. 
Note that we can compute the mean of $Y$ explicitly. 
By comparing a MC simulation with and without this control variate
(using $\alpha=1$) find the variance reduction from the control 
variate. Fishman find it is a factor of about $2.9$. 

\bigskip

\no 4. (partitioning) Consider the rainfall example from Owen's book,
section 8.5, chapter 8. In class I looked at the varaince reduction
from using proportional allocation (following Owen very closely).

\no (a). Reproduce this result.

\no (b). Now suppose we choose the sample sizes to be optimal. 
So $n_j$ is proportional to $p_j \sigma_j$ where $p_j$ is the probability 
of the $j$th strata and $\sigma_j^2$ is the variance within the $j$th 
strata. Normally you would do a preliminary MC to estimate the $\sigma_j$.
If you want to skip this you can find estimates of their values in Owen. 
With this optimal choice of the number of samples what is the variance 
reduction?

\bigskip

\no 5. (partitioning) Consider the network example. There is no obvious
way to partition. Come up with some partition and see how much 
variance reduction it gives. To begin with you can use proportional
allocation to choose the number of samples in each strata.
If you are more ambitious you can compute the variance within each
strata and use the optimal sample sizes for the strata. 

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