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title: 'Module 7: Supplementary exercises'
author: ""
date: ""
output: html_document
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## Exercise: Potential rejection sampling problems

First try to answer the following questions without using the computer -- then 
reuse the code from the supplementary slides to check your answer:

- Suppose we could not easily determine M and hence had to make a
conservative choice; say M = 100 or M = 500 in this context.
    1. Which effect will that have on the number of accepted samples?
    2. How would you have to compensate for a too large value of M if you want a
    given number of samples from the target distribution?
- What happens if you do not choose M large enough (e.g. M = 10 in our example)?
- What would be the effect of using a uniform proposal distribution on $[-10,10]$?
- What happens if the proposal distribution is an standard normal distribution (i.e. mean zero and standard deviation 1? Hints:
    1. You can use `dnorm()` for the normal density.
    2. You may have to create a sequence `x <- seq(-4, 4, by = 0.01)` to numerically evaluate the bound M relating `f0(x)` and `dnorm(x)`.

## Exercise: Improving the proposal distribution

If $f(x)$, $x\in[0,1]$ is a pdf on $[0,1]$ then for $a>0$, $1/a \cdot f(x/a)$, $x\in[0,a]$ is a pdf on $[0,a]$. Furthermore, a pdf on $[b, a+b]$ can be obtained
by simple translation.

- Based on these facts how can a beta distribution Be($\alpha$, $\beta$)
indirectly be used as the proposal distribution for our example?
-Implement the rejection sampling algorithm using Be(2.5,3.5) transformed to $[-4.1,4.1]$ (but with $M$ determined on $[-4,4]$).
- Check with a histogram that you are sampling the correct distribution.
- Find the acceptance rate.