05_03 - p 5-21  A special case of the gamma distribution occurs when = n/2 and  =1/2 for some positive integer n This is known as the

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: p. 5-21  A special case of the gamma distribution occurs when  = n /2 and  =1/2 for some positive integer n . This is known as the Chi-squared 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. 5-22  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. 5-23 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. 5-24  Summary for X ~ Beta(  ,  )  Pdf:  Cdf:  Parameters:...
View Full Document

This note was uploaded on 02/15/2012 for the course MATH 2810 taught by Professor Shao-weicheng during the Fall '11 term at National Tsing Hua University, China.

Page1 / 7

05_03 - p 5-21  A special case of the gamma distribution occurs when = n/2 and  =1/2 for some positive integer n This is known as the

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online