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Bernoulli Distribution

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BernoulliDistributionExample:Tossof coinDe neX= 1 if headcomesupandX= 0 if : (X= 1) = (X= 0) =12Examples:Often:Two outcomeswhich arenotequallylikely: Successof medicaltreatment Interviewed personis female Student passesexam Transmittanceof a diseaseBernoullidistribution(withparamet er ) Xtakes two values,0 and1, withprobabilitiespand1 p FrequencyfunctionofXp(x) = x(1 )1 xforx2f0;1g0otherwise Often:X= 1if eventAhasoccured0otherwiseExample:A= blood pressureabove 140 , Jan 30, 2003- 1 -BernoulliDistributionLetX1; : : : ; Xnbe independent Bernoullirandomvariableswithsameparamete r .FrequencyfunctionofX1; : : : ; Xnp(x1; : : : ; xn) =p(x1) p(xn) = x1+:::+xn(1 )n x1 ::: xnforxi2f0;1gandi= 1; : : : ; nExample:Paired-SampleSignTest Studysuccessof newelaboratesafety program Recordaverageweeklylossesin hoursof labor dueto accidents beforeandafterinstallationof theprogramin 10industrialplantsPlant12345678910Before 457346124335783342617After36604411935517 7292411De nefortheith plantXi= 1if rstvalueis greaterthanthesecond0otherwiseResult:111 1011111TheXi's areindependentlyBernoulli distributedwithunknownparameter.

– Frequency function of X p(x) = ‰ µx(1¡µ)1¡x for x 2 f0;1g 0 otherwise – Often: X = ‰ 1 if event A has occured 0 otherwise Example: A = blood pressure above 140/90 mm HG. Distributions, Jan 30, 2003 - 1 -

  Distribution, Frequency

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