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5102-Lecture-02 - Lecture 2 Stat 5102-004 21 January 2011...

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Lecture 2 Stat 5102-004 21 January 2011
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Parametric Families A family of distributions is a set whose elements are distributions. A parameter space is a set of indices for the distributions. Family Parameters Binomial p (0 , 1) Poisson λ (0 , ) Normal ( μ, σ 2 ) R × (0 , ) Typically, families are parameterized by distinguishing features of the distributions—means, variances, etc. STAT 5102 (Theory of Statistics) Lecture 2 1 / 81
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Binomial Y : n = 12 , p = 1 32 y 0 2 4 6 8 10 12 p 0.0 0.2 0.4 0.6 0.8 1.0 STAT 5102 (Theory of Statistics) Lecture 2 2 / 81
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Binomial Y : n = 12 , p = 1 16 y 0 2 4 6 8 10 12 p 0.0 0.2 0.4 0.6 0.8 1.0 STAT 5102 (Theory of Statistics) Lecture 2 3 / 81
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Binomial Y : n = 12 , p = 1 8 y 0 2 4 6 8 10 12 p 0.0 0.2 0.4 0.6 0.8 1.0 STAT 5102 (Theory of Statistics) Lecture 2 4 / 81
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Binomial Y : n = 12 , p = 1 4 y 0 2 4 6 8 10 12 p 0.0 0.2 0.4 0.6 0.8 1.0 STAT 5102 (Theory of Statistics) Lecture 2 5 / 81
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Binomial Y : n = 12 , p = 1 2 y 0 2 4 6 8 10 12 p 0.0 0.2 0.4 0.6 0.8 1.0 STAT 5102 (Theory of Statistics) Lecture 2 6 / 81
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Binomial Y : n = 12 , p = 3 4 y 0 2 4 6 8 10 12 p 0.0 0.2 0.4 0.6 0.8 1.0 STAT 5102 (Theory of Statistics) Lecture 2 7 / 81
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Binomial Y : n = 12 , p = 7 8 y 0 2 4 6 8 10 12 p 0.0 0.2 0.4 0.6 0.8 1.0 STAT 5102 (Theory of Statistics) Lecture 2 8 / 81
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Binomial Y : n = 12 , p = 15 16 y 0 2 4 6 8 10 12 p 0.0 0.2 0.4 0.6 0.8 1.0 STAT 5102 (Theory of Statistics) Lecture 2 9 / 81
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Binomial Y : n = 12 , p = 31 32 y 0 2 4 6 8 10 12 p 0.0 0.2 0.4 0.6 0.8 1.0 STAT 5102 (Theory of Statistics) Lecture 2 10 / 81
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Binomial Family Consider the binomial density f ( y | p ) = n y p 1 - p y (1 - p ) n y ∈ { 0 , . . . , n } p (0 , 1) . The parameter and observation values connect through the odds. g 1 : (0 , 1) (0 , ) p 7→ p 1 - p g - 1 2 : (0 , ) (0 , 1) ω 7→ ω 1 + ω Odds present a convenient scale for comparing probabilities. STAT 5102 (Theory of Statistics) Lecture 2 11 / 81
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Binomial Y : n = 12 , p = 1 32 odds = p 1 - p = 1 31 y 0 2 4 6 8 10 12 odds 0 5 10 15 20 25 30 STAT 5102 (Theory of Statistics) Lecture 2 12 / 81
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Binomial Y : n = 12 , p = 1 16 odds = p 1 - p = 1 15 y 0 2 4 6 8 10 12 odds 0 5 10 15 20 25 30 STAT 5102 (Theory of Statistics) Lecture 2 13 / 81
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Binomial Y : n = 12 , p = 1 8 odds = p 1 - p = 1 7 y 0 2 4 6 8 10 12 odds 0 5 10 15 20 25 30 STAT 5102 (Theory of Statistics) Lecture 2 14 / 81
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Binomial Y : n = 12 , p = 1 4 odds = p 1 - p = 1 3 y 0 2 4 6 8 10 12 odds 0 5 10 15 20 25 30 STAT 5102 (Theory of Statistics) Lecture 2 15 / 81
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Binomial Y : n = 12 , p = 1 2 odds = p 1 - p = 1 y 0 2 4 6 8 10 12 odds 0 5 10 15 20 25 30 STAT 5102 (Theory of Statistics) Lecture 2 16 / 81
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Binomial Y : n = 12 , p = 3 4 odds = p 1 - p = 3 y 0 2 4 6 8 10 12 odds 0 5 10 15 20 25 30 STAT 5102 (Theory of Statistics) Lecture 2 17 / 81
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Binomial Y : n = 12 , p = 7 8 odds = p 1 - p = 7 y 0 2 4 6 8 10 12 odds 0 5 10 15 20 25 30 STAT 5102 (Theory of Statistics) Lecture 2 18 / 81
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Binomial Y : n = 12 , p = 15 16 odds = p 1 - p = 15 y 0 2 4 6 8 10 12 odds 0 5 10 15 20 25 30 STAT 5102 (Theory of Statistics) Lecture 2 19 / 81
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