# Which distribution is correct in modeling conversion rate in a Monte Carlo

I am building a model for a Monte Carlo simulation that estimates the number of sales made for a door-to-door salesman.

Looking at his historic success by city, it seems he converts about 80% +/- 20% and the histogram looks like a bell curve.

From which distribution do I draw my "random conversion rate" to accurately reflect what I see? How did you determine the correct distribution?

I have tried using a normal distribution, but sometimes the random draw is larger than 1. I could "cap" it at 1, but it seems that there must be a better way!

• Are you merely trying to fit a parametric distribution to the data? You could try fitting a beta distribution, but what's the goal of the project? – dsaxton Jul 1 '15 at 20:49
• Two goals to the question : 1) how do I determine the proper distribution for factors going forward, 2) what distributions would be good candidates for this problem The goal of the project is to take a bunch of these variances into account and estimate the "sales per market" confidence interval - accuracy isn't super important, but I wanted to know the 'right' way to do this before I go using min(random.normalvariate(.80, .05), 1) as a 'good enough' metric – Robert Jul 1 '15 at 21:05
• "Monte Carlo" is not a statistical model, rather a statistical method. – Zhanxiong Jul 2 '15 at 12:31

Let's say you have $n$ classes with midpoints $m_i, i=1,\ldots,n$, relative frequencies $f_i, i=1,\ldots,n$, and breakpoints $0=b_0<b_1<b_2<\ldots<b_n=1$.
You can fit several versions of an empirical distribution, for instance using a piecewise linear approximation (linear on each interval $[b_{i-1},b_i]$).
Then you generate uniform points on $[0,1]$, observe which interval this belongs to and interpolate linearly.