# chapter3 - Discrete Random Variables and Probability...

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Unformatted text preview: Discrete Random Variables and Probability Distributions 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 y y F F F y p b Y P b F y Y P y F y p y y p y Y P y p b y y in increasing lly monotonica is ) ( 1 ) ( ) ( ) ( ) ( ) ( ) ( ) ( : (CDF) Function on Distributi Cumulative 1 ) ( ) ( ) ( ) ( : Function (Mass) y Probabilit all = ∞ =-∞ = ≤ = ≤ = = 2200 ≥ = = ∑ ∑-∞ = Example – Rolling 2 Dice (Red/Green) Red \ Green 1 2 3 4 5 6 1 2 3 4 5 6 7 2 3 4 5 6 7 8 3 4 5 6 7 8 9 4 5 6 7 8 9 10 5 6 7 8 9 10 11 6 7 8 9 10 11 12 Y = Sum of the up faces of the two die. Table gives value of y for all elements in S Rolling 2 Dice – Probability Mass Function & CDF y p(y) F(y) 2 1/36 1/36 3 2/36 3/36 4 3/36 6/36 5 4/36 10/36 6 5/36 15/36 7 6/36 21/36 8 5/36 26/36 9 4/36 30/36 10 3/36 33/36 11 2/36 35/36 12 1/36 36/36 ∑ = = = y t t p y F y y p 2 ) ( ) ( in result can die 2 ways of # to sum can die 2 ways of # ) ( Rolling 2 Dice – Probability Mass Function Dice Rolling Probability Function 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 2 3 4 5 6 7 8 9 10 11 12 y p(y) Rolling 2 Dice – Cumulative Distribution Function Dice Rolling - CDF 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 11 12 13 y F(y) Expected Values of Discrete RV’s • Mean (aka Expected Value) – Long-Run average value an RV (or function of RV) will take on • Variance – Average squared deviation between a realization of an RV (or function of RV) and its mean • Standard Deviation – Positive Square Root of Variance (in same units as the data) • Notation: – Mean: E( Y ) = μ – Variance: V( Y ) = σ 2 – Standard Deviation: σ Expected Values of Discrete RV’s [ ] [ ] [ ] ( 29 [ ] [ ] 2 2 2 2 2 all 2 all all 2 all 2 2 all 2 2 2 2 all all : Deviation Standard ) 1 ( ) ( 2 ) ( ) ( 2 ) ( ) ( 2 ) ( ) ( ) ( )) ( ( ) ( : Variance )...
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## This note was uploaded on 06/04/2011 for the course STA 4321 taught by Professor Staff during the Spring '08 term at University of Florida.

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chapter3 - Discrete Random Variables and Probability...

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