{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

02_Probability_part4

# 02_Probability_part4 - Ch1~6 p.49 3(TBp.156 If the moment...

This preview shows pages 1–4. Sign up to view the full content.

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 , and the 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 , V ar ( 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 , their joint mgf is de fi 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.

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

View Full Document
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 , X 2 , . . . , X n is φ X 1 X 2 ··· X n ( t 1 , t 2 , . . . , t 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 ) exists, 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 ) given 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 y p 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
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 .

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 12

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

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online