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05d STAT discrete random variables 4-APPENDICES

# 05d STAT discrete random variables 4-APPENDICES - 1...

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Unformatted text preview: 1 DISCRETE RANDOM VARIABLES Dr. V.R. Bencivenga Economic Statistics Economics 329 DISCRETE RANDOM VARIABLES — PART 3 OPTIONAL APPENDICES HISTORY OF THE POISSON DISTRIBUTION DERIVATION OF THE POISSON DISTRIBUTION TWO PROOFS FOR THE POISSON DISTRIBUTION MULTINOMIAL DISTRIBUTION NEGATIVE BINOMIAL (PASCAL) DISTRIBUTION GEOMETRIC DISTRIBUTION MOMENT GENERATING FUNCTIONS 2 DISCRETE RANDOM VARIABLES APPENDIX: HISTORY OF THE POISSON DISTRIBUTION 3 DISCRETE RANDOM VARIABLES History of the Poisson distribution : The Poisson distribution was derived by S.D. Poisson and published in 1837. He applied it to decisions of juries. It was not widely used until 1898, when von Bortkiewicz published Das Gesetz der Kleinen Zalhen , in which he applied the Poisson distribution to rare events. His most famous application was to the numbers of deaths in a Prussian army corp from being kicked by a horse. He had data for 14 army corps for 20 years (1875 to 1894, 280 observations). Below are the sample frequency distribution, and “expected frequencies,” based on the Poisson distribution: Number of deaths Observed frequency Expected frequency 0 144 139 1 91 97 2 32 34 3 11 8 4 2 1 5 and over 0 0 280 280 4 DISCRETE RANDOM VARIABLES APPENDIX: DERIVATION OF THE POISSON DISTRIBUTION 5 DISCRETE RANDOM VARIABLES Derivation of the Poisson distribution : The Poisson probability function is derived as the limit of the binomial probability function as n , holding np constant. The binomial probability function can be written as x n x x 1 i X ) p 1 ( p ! x ) 1 i n ( ) x ( P , since ! x ) 1 i n ( )! x n ( ! x ! n C x 1 i x n Substitute n p : x n x x x 1 i x n x x 1 i X n 1 n 1 ! x n ) 1 i n ( n 1 n ! x ) 1 i n ( ) x ( P 6 DISCRETE RANDOM VARIABLES Now take the limit, i.e, let n , while keeping constant x n x n n 1 n 1 ! x n 1 x n n 2 n n 1 n n n lim We will need the following lemma, which we will not prove: e n 1 lim n n It is easy to see the limit of each term and to get the result: ! x e n 1 n 1 ! x n 1 x n n 2 n n 1 n n n lim x 1 to goes x e to goes n x 1 to goes term each n...
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05d STAT discrete random variables 4-APPENDICES - 1...

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