Advanced Mathematical Methods Part II Statistics Probability Mel - - PowerPoint PPT Presentation

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Part II – Statistics Probability

Mel Slater

http://www.cs.ucl.ac.uk/staff/m.slater/Teaching/Statistics/

Advanced Mathematical Methods

Blaise Pascal

www.abarnett.demon.co.uk/atheism/wager.html

Pierre de Fermat Letters between these two in 17C about wagers led to foundations

• f probability theory

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Outline

Probability Axioms – Calculus of probability Conditional Probability and

Independence

Assigning Probabilities

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Probability

A measure of the degree of belief in

the truth of some proposition or the

• ccurrence of some event.

Probability theory is concerned with

how to find the probabilities of more complex events (propositions) given the probabilities of constituent simpler events.

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Domain or Sample Space

Probabilities are defined over a

domain of ‘elementary events’.

This is also called a sample space. Complex (compound) events are

formed out of the elementary events.

Mathematically the domain/sample

space is a set.

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Examples

Questionnaire (Virtual Reality Example):

To what extent did you have a sense of being in the place depicted by the virtual reality?

• Answer 1=Very Low … 7=Very High
• Domain = {1,2,3,…,7}

Two coins tossed: {HH,HT,TH,TT} Consider “It will rain tomorrow” – what is

the domain?

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Axioms of Probability

We use the term ‘this event

is true’ to mean that it has

• r will happen, and ‘false’
• therwise.

P(E) means ‘probability of

proposition E’.

http://kolmogorov.com/

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Axioms of Probability - Events

Examples – the questionnaire, the coin tossing

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Axioms of Probability

The triple … is the object of study: p is a

probability measure, which satisfies a number of axioms:

Such Ei are called ‘mutually exclusive’ – one and only one can be true

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Complements of Events

The event not-E is written as It is easy to show from the axioms that

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Conditional Probability

All probabilities are conditional

probabilities.

p(E|X) is the probability of E given

that we know X to be true.

Often the conditions are simply

understood, in which case p(E) can be used.

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Independence

Conditional probability is the final axiom: This leads to the definition of independent events: A and B are

independent if and only if: p(A|B) = p(A) and p(B|A) = p(B).

The complete definition for n events is: Where K can be any non-empty subset of {1,2,…,n}

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Probability of A or B

From the axioms the following can easily

be proved:

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Assigning Probabilities

Since probabilities are ‘subjective’ they

can be assigned however you like.

But some choices are more rational than

• thers.

Three methods are considered:

• Equal probabilities
• Betting odds
• Frequency

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Equal Probabilities

When there is no prior information over

the domain, every elementary event in the domain should have equal probability.

This is the case in classic ‘wagers’ such

as dice, coins, cards.

Eg, in 3 tosses of a coin the domain is

• {HHH,HHT,HTH,THH,

TTT,TTH,THT,HTT}

Each can be assigned probability 1/8.

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Betting Odds

Suppose that the betting odds on E

are a:b (or a/b:1).

This means that for every b pounds

you bet you get a if E occurs.

If you accept these odds then you

are agreeing that:

p(E) = b/(a+b). Alternatively p(not-E)/p(E) = a/b

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Fair Betting Odds

Betting odds are considered ‘fair’ when

each party expects on the average to gain 0.

• Suppose odds of x:1 are agreed for event E,

then p(E) = 1/(x+1).

• Suppose that probability reflects long run

frequency – i.e., that E happens 1/(x+1) of the time.

• Then 1/(x+1) of the time you will win x pounds

and x/(x+1) of the time you will lose 1 pound.

• Therefore your long run expected gain is

x/(x+1) – x/(x+1) = 0.

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Probability as Frequency

Suppose there are repeated trials of an

‘experiment’ in which E is a possible

• utcome.

The trials are all under equal conditions

and independent of one another.

Suppose the number of trials is n, and E

happens r times. Then

r/n → p(E) as n →∞ (with prob. 1)

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Summary

Probability measures degree of belief

• ver a domain.

The events or propositions that are

elements of the domain are elementary events.

However probabilities are assigned to

these elementary events the calculus of probability applies.

This is based on the axioms of probability Some terms to remember: mutually

exclusive, conditional, independence.