Chapter 8. Properties of Distributions

Chapter 8. Properties of Distributions - Â 8 Properties of...

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

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.

Unformatted text preview: Â§ 8 Properties of distributions Â§ 8.1 Expectation 8.1.1 Definition. Let X be a random variable. The expectation , (or expected value , or mean ) of X is: (i) (for discrete case) E [ X ] = X x âˆˆ X (Î©) x P ( X = x ); (ii) (for continuous case) E [ X ] = Z âˆž-âˆž xf ( x ) dx , where f is the pdf of X . 8.1.2 The expectation of X is what we would expect of its value if we are to take an observation of X . It is a weighted average of the whole range of values attainable by X (i.e. X (Î©)). More weight is given to points with higher mass/density function. 8.1.3 Proposition. Let X 1 ,...,X n be random variables. Define Y = g ( X 1 ,...,X n ), for some given real-valued function g . Then (i) (for discrete case) E [ Y ] = X x 1 Â·Â·Â· X x n g ( x 1 ,...,x n ) P ( X 1 = x 1 ,...,X n = x n ); (ii) (for continuous case) E [ Y ] = Z âˆž x 1 =-âˆž Â·Â·Â· Z âˆž x n =-âˆž g ( x 1 ,...,x n ) f ( x 1 ,...,x n ) dx n Â·Â·Â· dx 1 , where f is the joint pdf of ( X 1 ,...,X n ). This allows us to compute the expectation of any transformation of random variables using their joint distribution. 8.1.4 For constants Î±,Î² and random variables X,Y , E [ Î±X + Î² Y ] = Î± E [ X ] + Î² E [ Y ] . 8.1.5 X â‰¥ â‡’ E [ X ] â‰¥ (trivial!) 8.1.6 X â‰¥ Y â‡’ E [ X ] â‰¥ E [ Y ] (since X- Y â‰¥ .) 8.1.7 | E X | â‰¤ E | X | (since | X | â‰¥ both X and- X .) 47 8.1.8 If X = a on Î©, i.e. X is a non-random constant, then E [ X ] = a . (Reason â€” for continuous case, Z xf ( x ) dx = Z af ( x ) dx = a Z f ( x ) dx = a ; discrete case similar.) 8.1.9 X,Y independent random variables â‡’ E [ XY ] = E [ X ] E [ Y ] (Reason â€” for continuous case, Z Z xy f ( x,y ) dxdy = Z Z xy f X ( x ) f Y ( y ) dxdy = Z xf X ( x ) dx Â¶ Z y f Y ( y ) dy Â¶ ; discrete case similar.) 8.1.10 Examples. (i) X âˆ¼ Poisson ( Î» ): E [ X ] = âˆž X k =0 k P ( X = k ) = âˆž X k =1 k Î» k e- Î» k ! Â¶ = Î» âˆž X k =1 Î» k- 1 e- Î» ( k- 1)! = Î» âˆž X k =1 P ( X = k- 1) = Î» . (ii) X âˆ¼ geometric with success probability p : E [ X ] = âˆž X k =0 k (1- p ) k p =- p (1- p ) âˆž X k =0 d dp (1- p ) k =- p (1- p ) d dp 1 p Â¶ = 1- p p . (iii) X âˆ¼ negative binomial (no. of failures before k th success with success probability p ): Write X = X 1 + Â·Â·Â· + X k , where each X i âˆ¼ geometric with success probability p . Then E [ X ] = E [ X 1 ] + Â·Â·Â· + E [ X k ] = k (1- p ) p . (iv) N men and M women are seated randomly round a table. Let X = no. of men with a woman seated immediately to their right. Calculate E [ X ]. Solution : It is hard to find mass function of X explicitly. In fact, P ( X = k ) = N + M k N- 1 k- 1 Â¶ M- 1 k- 1 Â¶ / N + M N Â¶ . Easier... write X = X 1 + Â·Â·Â· + X N , where X i = ( 1 , i th man has a woman to his right, , otherwise....
View Full Document

This note was uploaded on 04/29/2010 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

Page1 / 13

Chapter 8. Properties of Distributions - Â 8 Properties of...

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

View Full Document
Ask a homework question - tutors are online