**This method allows operations to be performed between data distributions to simulate the resultant.**

## Introduction

Introduced by Stanislas Marcin Ulam and Nicholas Metropolis in 1949 to simulate phenomena in matter physics, this method uses X random data distributions to simulate the result of their addition, subtraction … It is called Monte Carlo in relation to the games of chance, roulette is one of the best known ways to obtain random numbers.

Mainly, it will be useful to simulate a **flow**, a level of **reliability**…

## 1 – Collect data from each of the variables

The first step is to collect the data of the subject that one represents in the form **histogram** to obtain distribution.

## 2 – Identify the mathematical model of our case

We define the mathematical model that makes the connection between the input variables and the resultant, in other words the equation Y = f(X).For this we will be in 3 cases :

- Either we are in a case where we have the formula beforehand. For example, we want to study the
**reliability**of a production line. We know that the formula of reliability is MTBF / (TMP + MTBF). We will collect the distribution of MTBF and TMP of each of our equipment of this line, to which we will apply this formula. These are the only cases where we will use the division et la substraction. - Either we are in a case where our data is independent. Clearly, the tasks are done in parallel. Without them having an interaction between her. We will then use the addition of variables . This is the case of the study of human size: it depends on the food, the DNA of our parents … which are variables independent of each other.
- Finally last case, our tasks are done in series in the sense that they are dependent each other. this is the typical case of a flow analysis, where the performance of machine 2 depends on the performance of machine 1. We will then use the multiplication of variables.

## 3 – Perform the calculation

The principle of calculating variable distribution is based on the fact that we want to calculate the set of possible probabilities of the resultant, based on the probabilities of the input variables. We want to calculate all the possible combinations of these variables.

For this reason, the calculation is done via a matrix, where the distribution frequencies of the first variable are put on the line and in the column, the distribution frequencies of the second variable. The crossing of these 2 elements representing our operation.

Note that for one :

**An Addition of variables :**we get a**Normal**distribution**Multiplication of variables :**we get a**Cauchy**distribution**Division of variables :**we get a**LogNormale**distribution**Substraction of variables :**we get a**Normal**distribution

## 4 – Conclude on the hypothesis

Once computed, we obtain a frequency table of the resultant of our model and thus the distribution of it. We can rightly perform the analyzes that we want of the type :

- I have a 50% chance to produce 20000 pieces a day
- The surface of my lake is 1000 m
^{2} - …