# Simulation with product of proportions in a biological process

I am modeling a biological process that uses plant tissue as the starting material to generate plants. The number of plants (k) successfully made with the inputs (n) is a small proportion (p). The purpose of the model is to set the inputs to guarantee attaining a certain minimum output with probability (P).

I have one year's worth of historical data. I started out modeling the process as a binomial distribution using the sample mean of observed proportions. However, the distribution of proportions is multi-modal and skewed. So I moved to a simulation-based approach.

In the simulation based approach, I fit the best distribution from a family of distributions to the sample of observed proportions. Then I sample this distribution, say, 10000 times. Each time, I use the set value of k to compute n. The cumulative distribution of n lets me set the input for a desired probability P.

Now, imagine the process is broken down into a sequence of steps, each with a characteristic success rate. For simplicity, assume two steps only. Thus, p = p_1 * p_2. Further, assume P represents a service-level that does not change. My question is about the sampling strategy to extend this simulation-based approach to multiple steps. When it comes to independent distributions that are chained, is sampling the product same as the product of samples?

Specifically, are the following sampling strategies equivalent to generate a representative sample of p from two independent distributions?

1. Take r pairs of p_1 and p_2 and take the product of each pair (See A)
2. Take r of p_1 and each time, take s of p_2 to make r*s products (See B)

The 2nd approach causes computation to explode when the number of sequential steps is large. if the first one is equivalent, I can use that. But intuition seems to suggest that if two independent random processes are chained, one must include the full universe of possibilities for consideration in math models.

I'd appreciate any help. • Maybe a strange question, but once you have your fitted distribution why do you simulate? I assume that the parametric model for it is known and you can simply determine the inverse of the CDF to get your desired value. – Jan Sep 4 '18 at 21:06
• You make a good point, Jan. I will try taking up the approach you have suggested. Thank you so much. – Sun Bee Sep 5 '18 at 2:09