Basics of probability calculus

1. By Simo Määttä, Kajaani University of appliedsciences, spring 2010 Basics of probabilitycalculus

2. 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

3. 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 % )

4. 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

5. 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 %

6. 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

7. 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)

8. 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.

9. 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) --

10. 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

11. example • Wecanpresentthisdistribution for examplebygraph as follows: This is a probabilitydistribution of ourrandomvariable X

12. Continuous and discreterandomvariable • Randomvariablesthatcanonlyhavesomespecificvaluearecalleddiscrete • Randomvariablesthatcanhaveeveryvaluebetweensomevaluesarecalledcontinuous

13. Continuousrandomvariables • In previousexamplerandomvariablewasdiscretebecauseitcouldonlyhavevalues 2, 3, …, 11 and 12 • How to describeprobabilitydistributions for continuousrandomvariablesthatcanhaveinfinitenumber of values???

14. 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

15. 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

16. Densityfunction • Example:Let X beperson’sheight

17. Densityfunction • Always: P(X getssomevalue) = 1

18. Densityfunction • Most common densityfunction is socallednormaldistribution.

19. Densityfunction • In thiscoursewewillneedChi-square –distribution.

20. 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