Session2

Session2 - EMSE 171/271: DATA ANALYSIS For Engineers and...

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EMSE 171/271: DATA ANALYSIS For Engineers and Scientists Session 2: Distribution Theory, Calculating Probabilities, Sums of Random Variables, Point-Estimation, Estimator Distributions Lecture Notes by: J. René van Dorp 1 www.seas.gwu.edu/~dorpjr 1 Department of Engineering Management and Systems Egineering, School of Engineering and Applied Science, The George Washington University, 1776 G Street, N.W. Suite 110, Washington ß D.C. 20052. E-mail: dorpjr@gwu.edu.
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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/18/05; ; Page 33 dorpjr@gwu.edu STATISTICAL REVIEW Distribution Theory Definition: The probability distribution or probability mass function (pmf) of a discrete rv is defined for every realization of by: \B \ :ÐBÑœT<Ð\œBÑœT<Ð+66 − À\Ð ÑœBÑ =H = Example 10: Consider a group of five potential blood donors and EßFßGßH IE F S of whom only and have type . Five blood samples, one from each individual, will be typed in random order until an individual is identified. S Define: the number of blood tests to identify an individual :Ð"Ñ œ T<Ð] œ "Ñ œ T<ÐE F œ œ Þ% or typed first) # & :Ð#Ñ œ T<Ð] œ #Ñ œ T<ÐGß H I E œ T<ÐGß H I ÑT<ÐE F Gß H I œœ & or first, and then or or first or next| or first) () () . 3 32 4
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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/18/05; ; Page 34 dorpjr@gwu.edu STATISTICAL REVIEW Distribution Theory :Ð Ñ œ T<Ð] œ Ñ œ Gß H I E œœ Þ # &$ 3 3 Pr( or first and second, and then or () 322 4 :Ð%Ñ œ T<Ð] œ %Ñ œ T<ÐGß H I œ œ Þ or all done first) ( )( )( ) 1 321 4 0 0.1 0.2 0.3 0.4 0.5 1234 y p(y) 0 0.1 0.2 0.3 0.4 0.5 012345 y p(y) A line graph of the pmf of Y A histogram graph of the pmf of Y y1234 p(y) 0.4 0.3 0.2 0.1 pmf of Y in Tabular form
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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/18/05; ; Page 35 dorpjr@gwu.edu STATISTICAL REVIEW Distribution Theory Definition: The cumulative distribution function (cdf) of a discrete rv with pmf \ :ÐBÑ B is defined for every number by: JÐBÑ œ T<Ð\ Ÿ BÑ œ :ÐCÑÞ ± CÀCŸB For any number , is the probability that the observed value of will be at BJÐB Ñ \ most . B When , :Ð+Ñ :Ð,Ñ  ! À T<Ð+\,ÑÁT<Ð+Ÿ\,ÑÁT<Ð+\Ÿ,ÑÁT<Ð+Ÿ\Ÿ,Ñ Example 11: Consider the blood typing example 10. y1234 p(y) 0.4 0.3 0.2 0.1 pmf of Y in Tabular form
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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/18/05; ; Page 36 dorpjr@gwu.edu STATISTICAL REVIEW Distribution Theory JÐ"Ñ œ T<Ð] Ÿ "Ñ œ T<Ð] œ "Ñ œ :Ð"Ñ œ Þ% JÐ#Ñ œ T<Ð] Ÿ #Ñ œ T<Ð] œ " #Ñ œ :Ð"Ñ  :Ð#Ñ œ %  Þ$ œ !Þ( JÐ$Ñ œ T<Ð] Ÿ $Ñ œ T<Ð] œ " # $Ñ œ :Ð"Ñ  :Ð#Ñ  :Ð$Ñ œ Þ(  !Þ# œ !Þ* JÐ% or . or or ÑœT<Ð] Ÿ%ÑœT<Ð] œ" # $ %Ñœ" or or or 0 0.2 0.4 0.6 0.8 1 012345 y F(y) 0.4 0.3 0.2 0.1 :ÐBÑ œ JÐBÑ  JÐB  "Ñ B œ "ß #ß $ß % ,
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EMSE 171/271 - FALL 2005 J.R. van Dorp - 10/18/05; ; Page 37 dorpjr@gwu.edu STATISTICAL REVIEW Distribution Theory Definition: probability distribution Let be a continuous rv. Then a or \ probability density function (pdf) of is a function such that for any two \0 Ð B Ñ numbers and with +, + Ÿ , ß T<Ð+Ÿ\Ÿ,Ñœ 0ÐBÑ.B ( + , Hence, equals the area under the density curve over the T<Ð+Ÿ\Ÿ,Ñ 0ÐBÑ interval . Ò+ß ,Ó x f(x) a b Note: T<Ð\œ+ÑœT<Ð+Ÿ\Ÿ+Ñœ! T<Ð+Ÿ\Ÿ,ÑœT<Ð+\Ÿ,ÑœT<Ð+Ÿ\,ÑœT<Ð+\,Ñ
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Session2 - EMSE 171/271: DATA ANALYSIS For Engineers and...

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