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Lecture 8 Probabilities and distributions

Lecture 8 Probabilities and distributions. Probability is the quotient of the number of desired events k through the total number of events n. If it is impossible to count k and n we might apply the stochastic definition of probability .

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Lecture 8 Probabilities and distributions

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  1. Lecture8 Probabilities and distributions Probabilityisthequotient of thenumber of desiredeventskthroughthetotalnumber of eventsn. Ifitisimpossible to count k and n we mightapplythestochasticdefinition of probability. Theprobability of an event j isapproximatelythefrequency of jduringnobservations.

  2. What is the probability to win in DużyLotek? We needthenumber of combinations of k events out of a total of N events Bernoulli distribution Thenumber of desiredeventsis 1. Thenumber of possibleeventscomesfromthenumber of combinations of 6 numbers out of 49.

  3. What is the probability to win in DużyLotek? We needtheprobabilitythat of a sample of Kelements out of a sampleuniverse of Nexactlynhave a desiredprobability and k not. Wrong! K=n+k n N Hypergeometric distribution P = 0.0186

  4. In Multi Lotek 20 numbers are taken out of a total of 80. What is the probability that you have exactly 10 numbers correct? N = 80 K = 20 n = 10 k = 10

  5. Assessingthenumber of infectedpersons Assessingtotalpopulationsize We take a sample of animals/plants and markthem We take a secondsample and countthenumber of markedindividuals Capture – recapturemethods Thefrequency of markedanimalsshouldequalthefrequencywothinthetotalpopulation Assumption: Closedpopulation Random catches Random dispersal Markedanimals do not differinbehaviour Nreal = 38

  6. Thetwosamplecase How many personshave a certaininfectuousdesease? Youtaketwosamples and countthenumber of infectedpersonsinthe first sample m1, inthesecondsample m2 and thenumber of infectedpersonsnotedinbothsamples k.

  7. In ecology we oftenhavethe problem to comparethespeciescomposition of twohabitats. Thespeciesoverlapismeasured by theSoerensendistancemetric. We do not knowwhether S islargeorsmall. To assesstheexpectation we construct a null model. Bothhabitatscontainspecies of a commonspeciespool. Ifthepoolsize n isknown we canestimatehow many joint species k containtwo random samples of size m and l out of n. Habitat B Habitat A k species l species m species n species Commonspeciespool Theprobability to getexactly k joint species. Probabilitydistribution. Theexpectednumber of joint species. Mathematicalexpectation

  8. Groundbeetlespecies of twopoplarplantations and twoadjacentwheetfields near Torun (Ulrich et al. 2004, AnnalesZool. Fenn.) Poolsize 90 to 110 species. Thereare much morespeciesincommonthanexpectedjust by chance. Theecologicalinterpretationisthatgroundbeetlescolonizefields and adjacentseminaturalhabitatsin a similarmanner. Groundbeetles do not colonizeaccording to ecologicalrequirements (niches) but according to spatialneighborhood.

  9. Bayesianinference and maximum likelihood (Idż na całość)

  10. The law of dependent propability Theorem of Bayes Abraham de Moivre (1667-1754) Thomas Bayes (1702-1761)

  11. Total probability A P(A|B3) P(A|B1) P(A|B2) B3 B2 B1 P(B2) P(B3) P(B1) N Idżnacałość Assume we choose gate 1 (G1) at the first choice. We are looking for the probability p(G1|M3) that the car is behind gate 1 if we know that the moderator opened gate 3 (M3).

  12. Calopteryx spelendens We study the occurrence of the damselfly Calopteryxsplendens at small rivers. We know from the literature that C. splendens occurs at about 10% of all rivers. Occurrence depends on water quality. Suppose we have five quality classes that occur in 10% (class I), 15% (class II), 27% (class III), 43% (class IV), and 5% (class V) of all rivers. The probability to find Calopteryx in these five classes is 1% (class I), 7% (class II), 14% (class III), 31% (class IV), and 47% (class V). To which class belongs probably a river if we find Calopteryx? p(class II|A) = 0.051, p(class III|A) = 0.183, p(class IV|A) = 0.647, p(class V|A) = 0.114 Indicator values

  13. Bayes and forensic Let’s take a standard DNA test for identifyingpersons. The test has a precision of more than 99%. Whatistheprobabilitythat we identifythewrong person? False positive fallacy Error of the prosecutor The forensic version of Bayes theorem

  14. Theerror of theadvocate In the process against the basketball star E. O. Simpson, one of his advocates (a Harvard professor) argued that Simpson sometimes has beaten his wife. However, only very few man who beat their wives later murder them (about 0.1%).

  15. Maximum likelihoods Suppose you studied 50 patients in a clinical trial and detected at 30 of them the presence of a certain bacterial disease. What is the most probable frequency of this disease in the population? We look for themaximumvalue of thelikelihoodfunction

  16. log likelihood estimator ln(Lp)

  17. Home work and literature • Refresh: • Probability • Permutations, variations, combinations • Bernoulli event • Pascal triangle, binomialcoefficients • Dependent probability • Independent probability • Derivative, integral of powerfunctions • Prepare to thenextlecture: • Arithmetic, geometric, harmonicmean • Cauchyinequality • Statisticaldistribution • Probabilitydistribution • Moments of distributions • Error law of Gauß Literature: http://www.brixtonhealth.com/CRCaseFinding.pdf

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