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.