5102-Lectures

5102-Lectures - Lecture 1 Stat 5102-004 19 January 2011...

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Lecture 1 Stat 5102-004 19 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 1 1 / 54
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Uniform Parameterizations Exercise Consider a family of continuous uniform distributions. The usual parameterization indexes the distributions by their endpoints. Y Unif( a , b ) f ( y | a , b ) = 1 b - a I ( a , b ) ( y ) y R Reparameterize the family according to midpoint and range. STAT 5102 (Theory of Statistics) Lecture 1 2 / 54
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Normal Y : E Y = 0 , Var Y = 1 y f(y) -6 -4 -2 0 2 4 6 0.0 0.4 μ σ 2 -6 -4 -2 0 2 4 6 0 1 3 4 STAT 5102 (Theory of Statistics) Lecture 1 3 / 54
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Normal Y : E Y = - 2 . 5 , Var Y = 1 y f(y) -6 -4 -2 0 2 4 6 0.0 0.4 μ σ 2 -6 -4 -2 0 2 4 6 0 1 3 4 STAT 5102 (Theory of Statistics) Lecture 1 4 / 54
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Normal Y : E Y = 2 . 5 , Var Y = 1 y f(y) -6 -4 -2 0 2 4 6 0.0 0.4 μ σ 2 -6 -4 -2 0 2 4 6 0 1 3 4 STAT 5102 (Theory of Statistics) Lecture 1 5 / 54
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Normal Location Family Translating a standard random variable Y 0 N (0 , 1) Y - 2 . 5 Y 0 - 2 . 5 Y 2 . 5 Y 0 + 2 . 5 forms a family of random variables Y μ Y 0 + μ μ ∈ {- 2 . 5 , 0 , 2 . 5 } distinguished by location. Y μ - μ N (0 , 1) μ ∈ {- 2 . 5 , 0 , 2 . 5 } STAT 5102 (Theory of Statistics) Lecture 1 6 / 54
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Normal Y : E Y = μ , Var Y = 1 , μ ∈ {- 2 . 5 , 0 , 2 . 5 } y f(y) -6 -4 -2 0 2 4 6 0.0 0.4 μ σ 2 -6 -4 -2 0 2 4 6 0 1 3 4 STAT 5102 (Theory of Statistics) Lecture 1 7 / 54
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Normal Y : E Y = 0 , Var Y = 1 y f(y) -6 -4 -2 0 2 4 6 0.0 0.4 μ σ 2 -6 -4 -2 0 2 4 6 0 1 3 4 STAT 5102 (Theory of Statistics) Lecture 1 8 / 54
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Normal Y : E Y = 0 , Var Y = 2 y f(y) -6 -4 -2 0 2 4 6 0.0 0.4 μ σ 2 -6 -4 -2 0 2 4 6 0 1 3 4 STAT 5102 (Theory of Statistics) Lecture 1 9 / 54
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Normal Y : E Y = 0 , Var Y = 3 y f(y) -6 -4 -2 0 2 4 6 0.0 0.4 μ σ 2 -6 -4 -2 0 2 4 6 0 1 3 4 STAT 5102 (Theory of Statistics) Lecture 1 10 / 54
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Normal Scale Family Rescaling a standard random variable Y 1 N (0 , 1) Y 2 2 Y 1 Y 3 3 Y 1 forms a family of random variables Y σ 2 σ Y 1 σ 2 ∈ { 1 , 2 , 3 } distinguished by scale. 1 σ Y σ 2 N (0 , 1) σ 2 ∈ { 1 , 2 , 3 } STAT 5102 (Theory of Statistics) Lecture 1 11 / 54
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Normal Y : E Y = 0 , Var Y = σ 2 , σ 2 ∈ { 1 , 2 , 3 } y f(y) -6 -4 -2 0 2 4 6 0.0 0.4 μ σ 2 -6 -4 -2 0 2 4 6 0 1 3 4 STAT 5102 (Theory of Statistics) Lecture 1 12 / 54
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This note was uploaded on 02/24/2011 for the course STAT 5102 taught by Professor Staff during the Spring '03 term at Minnesota.

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5102-Lectures - Lecture 1 Stat 5102-004 19 January 2011...

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