sta 2006 0809_lecturenotes_1

sta 2006 0809_lecturenotes_1 - • Mean • Variance •...

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Common Families of Distribution 1. Discrete Distribution (a) Uniform (b) Hypergeometric (c) Binomial (d) Poisson (e) Negative Binomial 2. Continuous Distribution (a) Uniform (b) Gamma (c) Normal (d) Beta (e) Exponential (f) Chi-square 3. Exponential Families A family of pdfs or pmfs is called an exponential family if it can be ex- pressed as f ( x ) = h ( x ) c ( θ ) exp( k i =1 w i ( θ ) t i ( x )) where h ( x ) , c ( θ ) 0, t i ( x ), independent of θ , and w i ( θ ), independent of x are real-valued functions. 1
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1. Discrete Distribution (a) Discrete Uniform Probability Mass Function Mean Variance Moment Generating Function 2
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(b) Hypergeometric Probability Mass Function Mean Variance Moment Generating Function 3
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(c) Binomial Probability Mass Function Mean Variance Moment Generating Function 4
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(d) Poisson Probability Mass Function Mean Variance Moment Generating Function 5
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(e) Negative Binomial Probability Mass Function Mean Variance Moment Generating Function 6
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2. Continuous Distribution (a) Uniform Probability Mass Function
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Unformatted text preview: • Mean • Variance • Moment Generating Function 7 (b) Gamma • Probability Mass Function • Mean • Variance • Moment Generating Function 8 (c) Normal • Probability Mass Function • Mean • Variance • Moment Generating Function 9 (d) Beta • Probability Mass Function • Mean • Variance • Moment Generating Function 10 (e) Exponential • Probability Mass Function • Mean • Variance • Moment Generating Function 11 (f) Chi-square • Probability Mass Function • Mean • Variance • Moment Generating Function 12 3. Exponential Families A family of pdfs or pmfs is called an exponential family if it can be ex-pressed as f ( x ) = h ( x ) c ( θ ) exp( k X i =1 w i ( θ ) t i ( x )) where h ( x ) , c ( θ ) ≥ 0, t i ( x ), independent of θ , and w i ( θ ), independent of x are real-valued functions. Examples • Binomial • Normal 13...
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