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NATCOR STOCHASTIC MODELLING (preliminary reading) Introduction to Probability & Random Variables IMPORTANT This material is designed to be viewed is two ways: In ‘Slide Show’ format, to take advantage of the animations; In ‘Notes Page’ format, so that you can see the accompanying notes. You might find it helps to print out the latter (to refer to) whilst viewing the former.

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Overview Concepts Laws and Notation Conditional Probability Total Law of Probability

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BASIC PROBABILITY CONCEPTS: ‘Experiments’ and ‘Events’ Experiment: an action whose outcome is uncertain (roll a die) Sample Space: set of all possible outcomes of an experiment (S = {1, 2, 3, 4, 5, 6}) Event: a subset of outcomes that is of interest to us (e.g. event 1 = even number, event 2 = higher than 4, event 3 = throw a 2, etc) Probability: measure of how likely an event is to occur (between 0 and 1)

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Classical Definition Classical Definition - Calculate, assuming events equally likely Relative Frequency Approach - Doing an experiment, using historical data Subjective/Bayesian Probability - Trusting instinct, using judgement

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Discrete Random Variables

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Probability Mass Function (pmf) of a discrete random variable

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Expected value and variance of discrete random variables

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Expected value and variance of combinations of discrete random variables

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Poisson Random Variable

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Poisson Random Variable

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Poisson Random Variable

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Continuous Random Variables

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Expected value and variance of continuous random variables

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Expected value and variance of combinations of continuous random variables NB. Exactly same as for discrete r.v.

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Exponential Random Variable

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Exponential Random Variable

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Exponential Example THEORY IF Events occur ‘at random’, at rate A per unit time. THEN: Time between events has an Exponential distribution, with mean of 1/A. AND: Time to next event has an Exponential distribution, with mean of 1/A.

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Normal Random Variable

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Standardised Normal Random Variable

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Normal Random Variables Important (in general) because: Many naturally occurring r.v.’s have a Normal distribution, e.g. weights, heights … Many useful statistics behave as Normal r.v.’s, even if the r.v.’s from which they derive are not Normal, e.g. Central Limit Theorem.