by simo m tt kajaani university of applied sciences spring 2010
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Basics of probability calculus

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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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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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