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Unformatted text preview: p. 521 A special case of the gamma distribution occurs when = n /2 and =1/2 for some positive integer n . This is known as the Chisquared distribution with n degrees of freedom (Chapter 6) Summary for X ~ Gamma( , ) Pdf: Cdf: Parameters: , Mean: E ( X ) = / Variance: Var ( X ) = / 2 . F ( x ) = γ ( α , λ x ) / Γ ( α ) . f ( x ) = ½ λ α Γ ( α ) x α − 1 e − λ x , if x ≥ , , if x < 0. • Beta Distribution Beta Function: For , > 0, the function f ( x ) = ( Γ ( α + β ) Γ ( α ) Γ ( β ) x α − 1 (1 − x ) β − 1 , if 0 ≤ x ≤ 1 , , otherwise, B ( α , β ) = R 1 x α − 1 (1 − x ) β − 1 dx = Γ ( α ) Γ ( β ) Γ ( α + β ) . is a pdf ( exercise ). p. 522 The distribution of a random variable X with this pdf is called the beta distribution with parameters and . The cdf of beta distribution can be expressed in terms of the incomplete beta function , i.e., F ( x )=0 for x <0, F ( x )=1 for x >1, and for 0 ≤ x ≤ 1, Theorem. The mean and variance of a beta distribution with parameters and are μ = α α + β and σ 2 = αβ ( α + β ) 2 ( α + β +1) . E ( X ) = R ∞ x Γ ( α + β ) Γ ( α ) Γ ( β ) x α − 1 (1 − x ) β − 1 dx = Γ ( α + β ) Γ ( α ) Γ ( β ) Γ ( α +1) Γ ( β ) Γ ( α + β +1) R ∞ Γ ( α + β +1) Γ ( α +1) Γ ( β ) x ( α +1) − 1 (1 − x ) β − 1 dx = α α + β . Proof. p. 523 E ( X 2 ) = R ∞ x 2 Γ ( α + β ) Γ ( α ) Γ ( β ) x α − 1 (1 − x ) β − 1 dx = Γ ( α + β ) Γ ( α ) Γ ( β ) Γ ( α +2) Γ ( β ) Γ ( α + β +2) R ∞ Γ ( α + β +2) Γ ( α +2) Γ ( β ) x ( α +2) − 1 (1 − x ) β − 1 dx = α ( α +1) ( α + β )( α + β +1) . Some properties When = =1, the beta distribution is the same as the uniform(0, 1). Whenever = , the beta distribution is symmetric about x =0.5, i.e., f (0.5 )= f (0.5+ ). As the common value of and increases, the distribution becomes more peaked at x =0.5 and there is less probability outside of the central portion. When > , values close to 0 become more likely than those close to 1; when < , values close to 1 are more likely than those close to 0 ( Q : How to connect it with E ( X )?) p. 524 Summary for X ~ Beta( , ) Pdf: Cdf: Parameters:...
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This note was uploaded on 02/15/2012 for the course MATH 2810 taught by Professor Shaoweicheng during the Fall '11 term at National Tsing Hua University, China.
 Fall '11
 ShaoWeiCheng
 Degrees Of Freedom, Probability

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