Basics of probability calculus

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# Basics of probability calculus - PowerPoint PPT Presentation

By Simo Määttä, Kajaani University of applied sciences , spring 2010. Basics of probability calculus. What is probability ?. Probability in general is a way to measure how certainly some outcome will happen in the future in some experiment .

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### Basics of probabilitycalculus

What is probability?
• Probability in general is a way to measure how certainly some outcome will happen in the future in some experiment.
• Experiment can be what ever situation that involves some kind of uncertainty
• Examples:
• 1. Throwing a coin
• 2. Measuring person’s height if person is selected randomly
Classicalprobability
• Wedefinethatelement is onepossibleoutcome in the experiment
• Complete group of allelement is calledsamplespace
• Classicalprobabilityassumesthatprobabilities of everyelementsareequal
• Example:

Throwing a dice.

Allelements (outcomes, 1, 2, 3, 4, 5 or 6) have the sameprobability of happening (1/6 *100% = 1,6666 % )

Denotions
• We denotegroups of possibleoutcomes as capital letter A, B, C and so on and wecallthemevents
• Example:Wethrough a dice. Possibleoutcomesare 1, 2, 3, 4, 5, 6. Nowwecould for examplehave A = {weget 2 or 3}
• Probability of A is denotedwith
• P(A) =probabilitythat A happens
• Probability of possibleevent A canbecalculated with formula
Example
• A = { Weget 1, 2 or 4}  P(A)=3/6 = 0,5  50%
• There is 7 men and 15 female in a class. What is the probabilitythatrandomlyselected person is a female?

Solution: 15 /(7+15) = 0,6818  68,2 %

Properties of probability
• Let A and B beevents
• Always 0≤P(A) ≤ 1
• If P(A) = 0 then A is impossibleevent
• If P(A) = 1 then A is 100 % certainevent
Properties of probability
• If A and B aremutuallyexclusivethen

P(A or B) = P(A) + P(B)

• WedenoteAc = complement of A (= eventwhere A willnothappen)

Now

P(Ac)= 1 – P(A)

• If G = group of allelements (possibleoutcomes) then
• P(G) = 1 (Probabilitythatsomethinghappens and obviouslysomethingalwayshappens)
Randomvariables
• Randomvariable is a functionthatattachnumber to each of the elements.
• Example:Person’sheight is a randomvariableif a person is selectedrandomly. Thisvariableattachnumber ”height in centimeters” to each person thatcanbeselected (elements)
• Wedenoterandomvariable as X, Y, etc.
Distribution of a randomvariable
• Certainvalues of randomvariablehavetheirownspecificprobabilities. Theseprobabilitiescanbepresentedwithprobabilitydistribution.

Example:

Wethroughtwodices. Letourrandomvariable X be the sum of the numbersweget. Nowelementsare (1, 1), (1, 2), (2, 1), …, (6, 6). X attachnumber 2 to outcome (1, 1) and number 3 to outcomes (1, 2) and (2, 1) and so on. Thereare 36 elements (possibleoutcomes) --

example
• Allelements and correspondingvalues of ourrandomvariableare in the followingtable:
• Nowwecancalculateprobablitities as

P(x=2) = 1/36, P(X=3)=2/36, … , P(X=12) = 1/36

example
• Wecanpresentthisdistribution for examplebygraph as follows:

This is a probabilitydistribution of ourrandomvariable X

Continuous and discreterandomvariable
• Randomvariablesthatcanonlyhavesomespecificvaluearecalleddiscrete
• Randomvariablesthatcanhaveeveryvaluebetweensomevaluesarecalledcontinuous
Continuousrandomvariables
• In previousexamplerandomvariablewasdiscretebecauseitcouldonlyhavevalues 2, 3, …, 11 and 12
• How to describeprobabilitydistributions for continuousrandomvariablesthatcanhaveinfinitenumber of values???
Densityfunction
• Probability of onespecificvalue of continuousrandomvariable is equal to 0!!!
• Example

Let X beweight.

Now P(X=45 kg) = 1/infinity = 0, becausethere is infinitenumber of possiblevalues for X (X is continuous) and 45 is onlyone of them.

•  wecan’tmakeprobabilitydistribution in a sameway as before
Densityfunction
• Probabilities of somevalues of continuousrandomvariable X canbecalculatedwith help of densityfunction(denotedwith f(x))
• Densityfunction f(x) is a graph in (x, y) coordinatesystem and itdescribesprobabilities of values of X
• Now P(x1<X<x2) = areabetweenx-axis and densityfunctioncurvebetweenvalues x1 and x2
Densityfunction
• Example:Let X beperson’sheight
Densityfunction
• Always: P(X getssomevalue) = 1
Densityfunction
• Most common densityfunction is socallednormaldistribution.
Densityfunction
• In thiscoursewewillneedChi-square –distribution.
Distributions
• Ifprobabilities of somecontinuousrandomvariableobeyssomedistributionwedenotethatwithsign ”~”
• Example
• X~ N(0,1), X obeysnormaldistributionwithmean of 0 and standarddeviation of 1
• X~χ2(f), X obeysChi-squaredistributionwithdegrees of freedom f (numberthatspecifies the shape of the distribution, seeearlierpicture). (χ is a greekletter)
• Probabilities of mostcommonlyuseddistributionshavebeentabulatedsowecangetdesiredprobabilitiesfromthesetables