ex-5: went on writing

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# Exercize 5 # Exercize 5
**Numerically compute an integral value via Monte Carlo approaches** The following integral must be evaluated:
The integral to be evaluated is the following:
$$ $$
I = \int\limits_0^1 dx \, e^x I = \int\limits_0^1 dx \, e^x
@ -143,7 +141,8 @@ For this reason, stratified sampling is used as a method of variance reduction
when MC methods are used to estimate population statistics from a known when MC methods are used to estimate population statistics from a known
population. population.
**MISER**
### MISER
The MISER technique aims to reduce the integration error through the use of The MISER technique aims to reduce the integration error through the use of
recursive stratified sampling. recursive stratified sampling.
@ -224,7 +223,95 @@ This time the error, altough it lies always in the same order of magnitude of
diff, seems to seesaw around the correct value. diff, seems to seesaw around the correct value.
## VEGAS \textcolor{red}{WIP} ## Importance sampling
In statistics, importance sampling is a technique for estimating properties of
a given distribution, while only having samples generated from a different
distribution than the distribution of interest.
Consider a sample of $n$ points {$x_i$} generated according to a probability
distribition function $P$ which gives thereby the following expected value:
$$
E [x, P] = \frac{1}{n} \sum_i x_i
$$
with variance:
$$
\sigma^2 [E, P] = \frac{\sigma^2 [x, P]}{n}
$$
where $i$ runs over the sample and $\sigma^2 [x, P]$ is the variance of the
sorted points.
The idea is to sample them from a different distribution to lower the variance
of $E[x, P]$. This is accomplished by choosing a random variable $y \geq 0$ such
that $E[y ,P] = 1$. Then, a new probability $P^{(y)}$ is defined in order to
satisfy:
$$
E [x, P] = E \left[ \frac{x}{y}, P^{(y)} \right]
$$
This new estimate is better then former one if:
$$
\sigma^2 \left[ \frac{x}{y}, P^{(y)} \right] < \sigma^2 [x, P]
$$
The best variable $y$ would be:
$$
y^{\star} = \frac{x}{E [x, P]} \thus \frac{x}{y^{\star}} = E [x, P]
$$
and a single sample under $P^{(y^{\star})}$ suffices to give its value.
---
The logic underlying importance sampling lies in a simple rearrangement of terms
in the integral to be computed:
$$
I = \int \limits_{\Omega} dx f(x) =
\int \limits_{\Omega} dx \, \frac{f(x)}{g(x)} \, g(x)=
\int \limits_{\Omega} dx \, w(x) \, g(x)
$$
where $w(x)$ is called 'importance function': a good importance function will be
large when the integrand is large and small otherwise.
---
For example, in some of these points the function value is lower compared to
others and therefore contributes less to the whole integral.
### VEGAS \textcolor{red}{WIP}
The VEGAS algorithm is based on importance sampling. It samples points from the The VEGAS algorithm is based on importance sampling. It samples points from the
probability distribution described by the function $f$, so that the points are probability distribution described by the function $f$, so that the points are