Probdists - Click to edit Master subtitle style Probability and Distributions A Brief Introduction Random Variables • Random Variable(RV A

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Unformatted text preview: Click to edit Master subtitle style 7/28/11 Probability and Distributions A Brief Introduction 7/28/11 Random Variables • Random Variable (RV): A numeric outcome that results from an experiment • For each element of an experiment’s sample space, the random variable can take on exactly one value • Discrete Random Variable: An RV that can take on only a finite or countably infinite set of outcomes • Continuous Random Variable: An RV that can take on any value along a continuum (but may be reported “discretely” • Random Variables are denoted by upper case letters ( Y ) 7/28/11 Probability Distributions • Probability Distribution: Table, Graph, or Formula that describes values a random variable can take on, and its corresponding probability (discrete RV) or density (continuous RV) • Discrete Probability Distribution: Assigns probabilities (masses) to the individual outcomes • Continuous Probability Distribution: Assigns density at individual points, probability of ranges can be obtained by integrating density function • Discrete Probabilities denoted by: p( y ) = P( Y=y ) • Continuous Densities denoted by: f( y ) 7/28/11 Discrete Probability Distributions Continuous Random Variables and Probability Distributions • Random Variable: Y • Cumulative Distribution Function (CDF): F ( y )=P( Y ≤ y ) • Probability Density Function (pdf): f ( y )=d F ( y )/d y • Rules governing continuous distributions: § f ( y ) ≥ 0 & y § 7/28/11 Expected Values of Continuous 7/28/11 Means and Variances of { } { } ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 1 2 1 1 1 2 2 1 1 1 1 1 constants random variables , 2 ,..., independent n i i i i i i i i i i j i i j j ij n n i i i i i i n n n n i i i i i j ij i i i j i n i i i U aY a Y E Y V Y COV Y Y E Y Y E U E aY a V U V aY a a a Y Y V U V aY μ σ μ μ σ μ σ σ = = =- = = = = + = ≡ ≡ = = =-- = ⇒ = = ⇒ = = + ≡ ⇒ = ∑ ∑ ∑ ∑ ∑ ∑ ∑ 2 2 1 1 n n i i i a σ = = = ∑ ∑ Normal (Gaussian) Distribution • Bell-shaped distribution with tendency for individuals to clump around the group median/mean • Used to model many biological phenomena • Many estimators have approximate normal sampling distributions (see Central Limit Obt aining Proba bilit ie s in EXCEL: To obt ain: F( y) = P( Y y) Use Funct ion: ≤ = NORM DIST( y, μ , σ ,1 ) Virt ually a ll st at ist ics t e xt books give t he cdf ( or uppe r t a il probabilit ie s) for st a ndardize d norm a l random variable s: z= ( y- μ ) / σ ~ N( 0 ,1 ) 7/28/11 Normal Distribution – Density Functions (pdf) 7/28/11 Intege r part and first decim al place of z Second Decimal Place of z 1-F(z) 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.39740....
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This note was uploaded on 07/28/2011 for the course STA 6208 taught by Professor Park during the Fall '08 term at University of Florida.

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Probdists - Click to edit Master subtitle style Probability and Distributions A Brief Introduction Random Variables • Random Variable(RV A

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