## 5.1 What does Monte Carlo simulation mean?

The definition of the Monte Carlo concept can be a bit confusing. For this reason, we will take Sawilowsky’s example and distinguish between: Simulation, Monte Carlo method and Monte Carlo simulation.

• A Simulation is a fictitious representation of reality. For example: Drawing one pseudo-random uniform variable from the interval [0,1] can be used to simulate the tossing of a coin. If the value is less than or equal to 0.50 designate the outcome as heads, but if the value is greater than 0.50 designate the outcome as tails. This is a simulation, but not a Monte Carlo simulation.

• A Monte Carlo method is a technique that can be used to solve a mathematical or statistical problem. For example: Pouring out a box of coins on a table, and then computing the ratio of coins that land heads versus tails is a Monte Carlo method of determining the behavior of repeated coin tosses, but it is not a simulation.

• A Monte Carlo simulation uses repeated sampling to obtain the statistical properties of some phenomenon (or behavior). For example: drawing a large number of pseudo-random uniform variables from the interval [0,1] at one time, or once at many different times, and assigning values less than or equal to 0.50 as heads and greater than 0.50 as tails, is a Monte Carlo simulation of the behavior of repeatedly tossing a coin.

The main idea behind this method is that a phenomenon is simulated multiple times on a computer using random-number generation based and the results are aggregated to provide statistical summaries associated to the phenomenon.

Sawilowsky lists the characteristics of a high-quality Monte Carlo simulation:

• the (pseudo-random) number generator has certain characteristics (e.g. a long “period” before the sequence repeats)

• the (pseudo-random) number generator produces values that pass tests for randomness

• there are enough samples to ensure accurate results

• the algorithm used is valid for what is being modeled

• it simulates the phenomenon in question.