Stochastic simulation

Key takeaways

Stochastic simulation belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Stochastic simulation to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Stochastic simulation from memory before moving on to harder problems.

Reference excerpt

A stochastic simulation is a simulation of a system that has variables that can change stochastically (randomly) with individual probabilities. Realizations of these random variables are generated and inserted into a model of the system. Outputs of the model are recorded, and then the process is repeated with a new set of random values. These steps are repeated until a sufficient amount of data is gathered. In the end, the distribution of the outputs shows the most probable estimates as well as a frame of expectations regarding what ranges of values the variables are more or less likely to fall in. Often random variables inserted into the model are created on a computer with a random number generator (RNG). The U(0,1) uniform distribution outputs of the random number generator are then transformed into random variables with probability distributions that are used in the system model.

Etymology Stochastic originally meant "pertaining to conjecture"; from Greek stokhastikos "able to guess, conjecturing": from stokhazesthai "guess"; from stokhos "a guess, aim, target, mark". The sense of "randomly determined" was first recorded in 1934, from German Stochastik.

Discrete-event simulation In order to determine the next event in a stochastic simulation, the rates of all possible changes to the state of the model are computed, and then ordered in an array. Next, the cumulative sum of the array is taken, and the final cell contains the number R, where R is the total event rate. This cumulative array is now a discrete cumulative distribution, and can be used to choose the next event by picking a random number z~U(0,R) and choosing the first event, such that z is less than the rate associated with that event.

Probability distributions A probability distribution is used to describe the potential outcome of a random variable. Limits the outcomes where the variable can only take on discrete values.

Bernoulli distribution

A random variable X is Bernoulli-distributed with parameter p if it has two possible outcomes usually encoded 1 (success or default) or 0 (failure or survival) where the probabilities of success and failure are P ( X = 1 ) = p {\displaystyle P(X=1)=p} and P ( X = 0 ) = 1 − p {\displaystyle P(X=0)=1-p} where 0 ≤ p ≤ 1 {\displaystyle 0\leq p\leq 1} . To produce a random variable X with a Bernoulli distribution from a U(0,1) uniform distribution made by a random number generator, we define

X = { 1 , if 0 ≤ U < p 0 , if 1 ≥ U ≥ p {\displaystyle X={\begin{cases}1,&{\text{if }}0\leq U<p\\0,&{\text{if }}1\geq U\geq p\end{cases}}}

such that the probability for P ( X = 1 ) = P ( 0 ≤ U < p ) = p {\displaystyle P(X=1)=P(0\leq U<p)=p} and P ( X = 0 ) = P ( 1 ≥ U ≥ p ) = 1 − p {\displaystyle P(X=0)=P(1\geq U\geq p)=1-p} .

Example: Toss of coin Define

X = { 1 if heads comes up 0 if tails comes up {\displaystyle X={\begin{cases}1&{\text{if heads comes up}}\\0&{\text{if tails comes up}}\end{cases}}}

For a fair coin, both realizations are equally likely. We can generate realizations of this random variable X from a U ( 1 , 0 ) {\displaystyle U(1,0)} uniform distribution provided by a random number generator (RNG) by having X = 1 {\displaystyle X=1} if the RNG outputs a value between 0 and 0.5 and X = 0 {\displaystyle X=0} if the RNG outputs a value between 0.5 and 1.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Stochastic simulation

Start with the simplest possible case. Write down what Stochastic simulation claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Stochastic simulation before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Stochastic simulation ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Stochastic simulation

In research: Stochastic simulation appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Stochastic simulation in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Stochastic simulation is common in secondary-school and first-year university syllabi. It links to neighbouring topics Stochastic processes, Stochastic simulation, so understanding it makes those chapters shorter.
In everyday life: Look for Stochastic simulation outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Stochastic simulation in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Stochastic simulation means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Stochastic simulation out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Stochastic simulation in simple terms?

A stochastic simulation is a simulation of a system that has variables that can change stochastically (randomly) with individual probabilities. Realizations of these random variables are generated and inserted into a model of the system.

Why does Stochastic simulation matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Stochastic simulation?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Stochastic simulation.