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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

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