Lecture 13 - Probability Distribution Functions & Monte Carlo

Addressing Uncertainty

We address uncertainties in our models, more speicfically in our parameters, by applying the use of probability distributions and Monte Carlow methods. This adds stochasticity to the model which will allow the model to produce a range of possible outcomes rather than one definite one.

• This is better at representing real world systems due to how they tend to be random

Random Variables

Have a single value that is determined by chance for each iteration of a trial.

They can be discrete, taking a countable number of distinct discrete values (integers, heads or tails)

Or continuous, taking an infinite number of possible values (numerical, concentration, height, time, etc)

Probability Distributions

A way to describe how likely each outcome of an uncertain event is, allows us to account for uncertainties. These outcomes include thigns such as an outcome, parameters, and so on. Depending on what we want to model, we’d use different probability distirbutions to represent it.

Bernoulli Distribution

Used for discrete random variables when there are two possibilities (random variable is either 0 or 1, etc)

Binomial Distribution

Used for discrete random variables when we want to figure out the probability of observing a certain number of outcomes/success in N trials (probability number of 4 heads in 10 coin tosses)

Poisson Distribution

Use for the probability of a number of events occuring in a fixed time with a known average rate, used for rare events

Uniform Distribution

Used for discrete and continuous random variables, when there is a specific range (with min and max) of an outcome that all have the same possibilities, there is no preferred value

Normal/Gaussian Distribution

Used for parameters that follow a normal distribution, when a certain value (the mean) is more probable than others

Monte Carlo Methods

Involves running a model many times (hundreds of thousands) where each time it is ran, the parameters with a probability distribution takes a random deviate. Each model parameter will have a defined distribution type (based on the ones mentioned prior).

• They may also have bounds if necessary

The user will then select number of model runs (n) to be conducted and for each instance, each parameter will generate a deviate (indepdently), which is recorded, and the outcome is saved

• We’d also typically run an initial analytical run where we just use the mean of the probability distirbution for each parameter as a comparison baseline

After running it n amount of times, we’d take all the outcomes and find the summary statistics of it

• This tells us the ranges of possible values with which ones are more probable