02_Probability_part4

02_Probability_part4 - Ch1~6, p.49 3. (TBp.156) If the...

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NTHU MATH 2820, 2008, Lecture Notes Ch1~6, p.49 Definition 3.4 (moment, TBp. 155) Proof. M ( k ) X (0) = d k dt k M X ( t ) t =0 = d k dt k −∞ e tx dF X ( x ) t =0 = −∞ d k dt k e tx t =0 dF X ( x )= −∞ x k e tx t =0 dF X ( x ) = −∞ x k · 1 dF X ( x E ( X k ) . Proof. Proof. M a + bX ( t E [ e t ( a + bX ) ]= e at E [ e ( bt ) X e at M X ( bt ) . M X + Y ( t E [ e t ( X + Y ) E ( e tX e tY ) = E ( e tX ) E ( e tY M X ( t ) M Y ( t ) . 3. (TBp.156) If the moment generating function exists in an open interval containing zero, then M ( k ) X (0) = E ( X k ) . 4. (TBp.158) For any constants a,b , M a + bX ( t e at M X ( bt ) . 5. (TBp.159) X , Y independent M X + Y ( t M X ( t ) M Y ( t ) . 6. continuity theorem (see Chapter 5) The k th moment of a random variable is E ( X k ) μ k ,andthe k th central moment is E [( X μ X ) k ] μ 0 k . made by Shao-Wei Cheng (NTHU, Taiwan) Ch1~6, p.50 ¾ Some Notes. ± μ k = E ( X k E { [( X μ X )+ μ X ] k } = k i =0 k i ( μ X ) n i E [( X μ X ) i ] = k i =0 k i ( μ X ) n i μ 0 i . In particular, E ( X μ X = μ 1 , and , Var ( X σ 2 X = μ 2 μ 2 1 = μ 0 2 . Definition 3.5 (joint moment generating function, TBp. 161) ± ± μ 0 k = E [( X μ X ) k E k i =0 k i ( μ X ) n i X i = k i =0 k i ( μ X ) n i E ( X i ) = k i =0 k i ( μ X ) n i μ i . For random variables X 1 ,X 2 ,...,X n ,the ir joint mgf is de f ned as: M X 1 X 2 ··· X n ( t 1 ,t 2 ,...,t n E ( e t 1 X 1 + t 2 X 2 + ··· + t n X n ) if the expection exists.
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NTHU MATH 2820, 2008, Lecture Notes Ch1~6, p.51 Theorem 3.6 (properties of joint mgf) 1. M X 1 ( t 1 )= M X 1 X 2 ··· X n ( t 1 , 0 ,..., 0) 2. uniqueness theorem 3. X 1 ,X 2 ,...,X n are independent if and only if M X 1 X 2 ··· X n ( t 1 ,t 2 ,...,t n n i =1 M X i ( t i ) . 4. r 1 + ··· + r n t r 1 1 ··· t r n n M X 1 X 2 ··· X n (0 , 0 0) = E ( X r 1 1 X r 2 2 X r n n ) Definition 3.6 (characteristic function, TBp. 161) The characteristic function (chf) of a random variable X is φ X ( t E ( e itX ) , where i = 1, and the joint characteristic function of X 1 2 n is φ X 1 X 2 ··· X n ( t 1 2 n E ( e it 1 X 1 + it 2 X 2 + ··· + it n X n ) . made by Shao-Wei Cheng (NTHU, Taiwan) Ch1~6, p.52 Theorem 3.7 (properties of characteristic function) • conditional expectation 1. The characteristic function always exists. 2. If M X ( t )ex ists ,then φ X ( t M X ( it ). 3. uniqueness theorem 4. The properties of characteristic function are similar to those of moment generating function. Definition 3.7 (conditional expection, TBp. 135-136) The conditional expectation of h ( Y )g iven X = x is [Discrete case] : E ( h ( Y ) | X = x y h ( y ) p Y | X ( y | x ) In particular, E ( Y | X = x y yp Y | X ( y | x ) [Continuous case] : E ( h ( Y ) | X = x h ( y ) f Y | X ( y | x ) dy In particular, E ( Y | X = x yf Y | X ( y | x ) dy
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NTHU MATH 2820, 2008, Lecture Notes Ch1~6, p.53 X Y f ( x , y ): joint pdf ¾ f ( x , y ): a joint pdf. ¾ Fix x * , is f ( x * , y ) a pdf of y ? ¾ f Y | X ( y | x * )= f ( x * , y )/ f X ( x * ) is a pdf of y since f X ( x )= −∞ f ( x ,y ) dy =1? −∞ f ( x ,y ) dy f X ( x ) =1 . ¾ E ( Y | x * ): mean of f Y | X ( y | x * ). ¾ Example. X =age (unit=year), Y =height (unit=cm) ± Y | X = x : a random variable (unit=cm) that represents the height distribution of people with age= x .
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This note was uploaded on 03/11/2011 for the course STA 506 taught by Professor Lisa during the Spring '11 term at West Chester.

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02_Probability_part4 - Ch1~6, p.49 3. (TBp.156) If the...

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