Probdists - Probability and Distributions A Brief...

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Unformatted text preview: Probability and Distributions A Brief Introduction 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 ) • Individual outcomes for RV are denoted by lower case letters ( y ) 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 ) • Cumulative Distribution Function: F( y ) = P( Y y ≤ ) Discrete Probability Distributions all Probability (Mass) Function: ( ) ( ) ( ) ( ) 1 Cumulative Distribution Function (CDF): ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) is monotonically increasing in y b y p y P Y y p y y p y F y P Y y F b P Y b p y F F F y y =-∞ = = ≥ 2200 = = ≤ = ≤ =-∞ = ∞ = ∑ ∑ 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 ≥ 2200 y P(a ≤ Y b) = ≤ F (b)- F (a) = P( Y =a) = 0 2200 a ∫ b a dy y f ) ( 1 ) ( = ∫ ∞ ∞- dy y f Expected Values of Continuous RVs [ ] [ ] ( 29 [ ] [ ] [ ] [ ] ( 29 [ ] ( 29 σ σ σ μ μ μ μ μ μ μ μ μ μ μ μ μ μ σ μ a a Y V a dy y f y a dy y f a ay dy y f b a b ay b aY E b aY E b aY V b a b a dy y f b dy y yf a dy y f b ay b aY E Y E Y E dy y f dy y yf dy y f y dy y f y y dy y f y Y E Y E Y V dy y f y g Y g E dy y yf Y E b aY = = =- =- = = +- + = +- + = + + = + = = + = + = +- = +- = = +- = +- = =- =- = = = = = + ∞ ∞- ∞ ∞- ∞ ∞- ∞ ∞- ∞ ∞- ∞ ∞- ∞ ∞- ∞ ∞- ∞ ∞- ∞ ∞- ∞ ∞- ∞ ∞- ∞ ∞- ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) ( ) ( ) ( ) 1 ( ) ( 2 ) ( ) ( 2 ) ( ) ( 2 ) ( ) ( )) ( ( ) ( : Variance ) ( ) ( ) ( e) convergenc absolute (assuming ) ( ) ( : Value Expected Means and Variances of Linear Functions of RVs { } { } ( 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...
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Probdists - Probability and Distributions A Brief...

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