Usbrth

I've read the following (here on page 2):

Suppose that you want to approximate a function $f$. One way to do this is to produce a sampling distribution proportional to $f$ and then make a histogram of samples taken from the distribution. The resulting histogram will be proportional to $f$ (obviously), so it only needs to be scaled to approximate $f$.

The procedure can be summarized as follows:

• Create a sampling distribution proportial to $f$
  


• Make a histogram of samples taken from the sampling distribution


• Scale the histogram to approximate $f$
  

The sacle factor $s$ needed to make the histogram approximate $f$ is the ratio of the average value $v$ of $f$ over the sampling domain to the average number $h$ of samples per bin in the histogram, i.e. $s=v/h$.

I'm not sure how seriously this has to be taken, but could anybody explain to me (in a more formal way) what the author is meaning to say?

Let's consider a example: Assume $f$ is the density of the standard normal distribution $mathcal N_0,:1$. We could divide an interval $[a,b]$ into $C$ "bins" of size $delta$. Now we could draw $n$ samples from $mathcal N_0,:1$ and record for each bin $i$ the number $B(i)$ of samples falling into that bin (if $xin[a,b)$ is a sample, it lies in the $lfloorfracx-adeltarfloor$-th bin).

Clearly, $$[a,b)ni xmapsto Bleft(lfloorfracx-adeltarfloorright)tag1$$ is an approximation of the shape of $f$.

Now, let $v$ be the average value of $f$ on $[a,b]$, $h$ be the average number of samples per bin and $s:=v/h$. If I got it right, the desired approximation would be $$tilde f(x):=sBleft(lfloorfracx-adeltarfloorright);;;textfor xin[a,b).$$ Here's a plot of the result for $a=-5$, $b=5$, $C=2000$, $delta=(b-a)/C$ and $n=1000000$:

Obviously, the scale is not correct. Did I made any mistake or is there something wrong with the description in the paper?