02_Probability_part2

02_Probability_part2 - Chapters 2 and 3 Ch1~6, p.9 Outline...

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NTHU MATH 2820, 2008, Lecture Notes Ch1~6, p.9 Chapters 2 and 3 Definition 2.1 (random variable, TBp. 33) random variable ¾ conditional distribution ¾ independent random variables ¾ function of random variables • distribution of transformed r.v. • extrema and order statistics ¾ random variables ¾ distribution •discrete and continuous •univariate and multivariate •cdf, pmf, pdf Outline A random variable is a function from to the real numbers. R R R L L made by Shao-Wei Cheng (NTHU, Taiwan) Ch1~6, p.10 Example 2.1 (cont. Ex. 1.1) Question 2.1 Why statisticians need random variables? Why they map to real line? (1) X 1 = the total number of heads (2) X 2 = the number of heads on the f rst toss (3) X 3 = the number of heads minus the number of tails = { hhh,hht,hth,thh,htt, tht,tth,ttt } 1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8
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Ch1~6, p.11 • distribution Question 2.2 A random variable have a sample space on real line . Does it bring some special ways to describe its probability measure? • joint pdf • joint cdf • joint mgf/cf • joint pmf • joint cdf • joint mgf/cf multi- variate r.v.’s •pdf •cdf •mgf/cf •pmf •cdf •mgf/cf uni- variate r.v. continuous discrete mgf (moment generating function) and cf (characteristic function) will be defined in Chapter 4 pmf: probability mass function, pdf: probability density function, cdf: cumulative distribution function made by Shao-Wei Cheng (NTHU, Taiwan) Ch1~6, p.12 Definition 2.2 (discrete and continuous random variables, TBp. 35 and 47) Definition 2.3 (cumulative distribution function, TBp. 36) A discrete random variable cantakeonon lya f nite or at most a countably in f nite number of values. A continuous ran- dom variable can take on a continuum of values. Afunct ion F is called the cumulative distribution function (cdf) of a random variable X if F ( x )= P ( X x ) ,x R . R Q : Does data following a continuous distribution
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02_Probability_part2 - Chapters 2 and 3 Ch1~6, p.9 Outline...

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