= E{ E[ | ] (Y) } X2 Y h2 = E{ } E { (Y) } X2 h2 This thinks of in the context of above. X2 g(X) Next examine Var{ E( | ) }. X Y Var{ E( | ) } = Var{ } = E { (Y) } [ E( ) ]2 X Y h(Y) h2 h(Y) Of course E( ) = E( E(X | ) ) = E(X). h(Y) Y = E { (Y) } [ E(X) ]2 = E { (Y) } h2 h2 Interesting. Now add the last two results: E{ Var (X | ) } + Var{ E( | ) } Y X Y = E{ } E { (Y) } + E { (Y) } X2 h2 h2 = E{ } = Var(X) X2 This proves the incredibly insightful result that Var(X) = E{ Var (X | ) } + Var{ E( | ) } Y X Y As a particular useful consequence Var(X) Var{ E( | ) }. This is the X Y ≥ essence of the Rao-Blackwell theorem. Let’s next consider the definition of the For any positive value we gamma function. q define the gamma function of as the integral q (q) = As a aside, let’s note that Rice, page 53, defines this as (x) = In this form, there are two dummy variables. First of all, is the dummy variable used to u define the integral, and it’s used in the integral. This could be replaced by any other only letter (assuming no conflicts with other letters) without changing the meaning. Secondly, is the dummy variable used to define the function. The letter appears on both sides of x x the equals sign. It could also be replaced with any other (nonconflicting) letter. Thus, the two definitions of the gamma function just provided are identical.
= E{ E[ X2 | Y ] h2 (Y) } = E{ X2 } E { h2 (Y) } This thinks of X2 in the context of g(X) above. Next examine Var{ E( X | Y ) }. Var{ E( X | Y ) } = Var{ h(Y) } = E { h2 (Y) } [ E( h(Y) ) ]2 Of course E( h(Y) ) = E( E(X | Y ) ) = E(X). = E { h2 (Y) } [ E(X) ]2 = E { h2 (Y) } Interesting. Now add the last two results: E{ Var (X | Y ) } + Var{ E( X | Y ) } = E{ X2 } E { h2 (Y) } + E { h2 (Y) } = E{ X2 } = Var(X) This proves the incredibly insightful result that Var(X) = E{ Var (X | Y ) } + Var{ E( X | Y ) } As a particular useful consequence Var(X) ≥ Var{ E( X | Y ) }. This is the essence of the Rao-Blackwell theorem. Let’s next consider the definition of the gamma function. For any positive value q we define the gamma function of q as the integral (q) = As a aside, let’s note that Rice, page 53, defines this as (x) = In this form, there are two dummy variables. First of all, u is the dummy variable used to define the integral, and it’s used only in the integral. This could be replaced by any other letter (assuming no conflicts with other letters) without changing the meaning. Secondly, x is the dummy variable used to define the function. The letter x appears on both sides of the equals sign. It could also be replaced with any other (nonconflicting) letter. Thus, the two definitions of the gamma function just provided are identical.
NOTES2011FEB09.docx