By simo m tt kajaani university of applied sciences spring 2010
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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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By simo m tt kajaani university of applied sciences spring 2010

By Simo Määttä, Kajaani University of appliedsciences, spring 2010

Basics of probabilitycalculus


What is probability
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


Classical probability
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
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
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
    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 probability1
    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)


    Random variables
    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 random variable
    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) --


    Example1
    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


    Example2
    example

    • Wecanpresentthisdistribution for examplebygraph as follows:

    This is a probabilitydistribution of ourrandomvariable X


    Continuous and discrete random variable
    Continuous and discreterandomvariable

    • Randomvariablesthatcanonlyhavesomespecificvaluearecalleddiscrete

    • Randomvariablesthatcanhaveeveryvaluebetweensomevaluesarecalledcontinuous


    Continuous random variables
    Continuousrandomvariables

    • In previousexamplerandomvariablewasdiscretebecauseitcouldonlyhavevalues 2, 3, …, 11 and 12

    • How to describeprobabilitydistributions for continuousrandomvariablesthatcanhaveinfinitenumber of values???


    Density function
    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


    Density function1
    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


    Density function2
    Densityfunction

    • Example:Let X beperson’sheight


    Density function3
    Densityfunction

    • Always: P(X getssomevalue) = 1


    Density function4
    Densityfunction

    • Most common densityfunction is socallednormaldistribution.


    Density function5
    Densityfunction

    • In thiscoursewewillneedChi-square –distribution.


    Distributions
    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


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