ρ < 1 be five real numbers. The

Transcription

1 Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace Chapter 4 Multple Radom Varables Chapter 4.5 Covarace ad Correlato Theorem 4.5.7: For ay radom varables ad, a. ρ. b. ρ = f ad oly f there exst umbers a 0 ad bsuch that P ( = a+ b) =. If ρ =, the a > 0, ad f ρ =, the a < 0. See Example (p ) for a example where ad are strogly related (o-lear) but ther covarace s 0. Defto 4.5.0: Let < µ <, < µ <, σ > 0, σ > 0, ad < ρ < be fve real umbers. The bvarate ormal pdf wth meas µ ad µ, varaces σ ad σ, ad correlato ρ s the bvarate pdf gve by

2 Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 x µ x µ y µ y µ f( x, y) = exp ( ) ρ( )( ) + ( ) πσ σ for < x <, < y <. ρ ( ρ ) σ σ σ σ, Alteratvely, we ca re-wrte ths formula a geeral way. Defe ρσ σ, whch the varace- covarace of the bvarate ormal dstrbuto, the we have σ Σ= ρσ σ σ T f( x, y) = exp (, ) (, ) / x µ y µ Σ x µ y µ ( π ) Σ. Propertes:. The margal dstrbutos of ad are µ σ ad. The correlato betwee ad s ρ = ρ. (, ) µ σ (, ), respectvely. 3. For ay costats a ad b, the dstrbuto of a + b s a + b a + b + ab. ( µ µ, σ σ ρσ σ) 4. The codtoal dstrbutos of gve = y ad of gve = x are also ormal dstrbutos. The pdf of = y s ( µ ρσ ( / σ )( y µ ), σ ( ρ )) ad are depedet f ad oly f ρ = 0. Example (Margal Normalty Does ot Imply Jot Normalty): Let ad are depedet radom varables wth the pdf (0,), defe

3 Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006, f > 0 Z = -, f < 0. The Z has ormal dstrbuto but Z ad s ot bvarate ormal. Chapter 4.6 Multvarate Dstrbutos Cosder a radom vector =(,,, ) wth sample space that s a subset of x = ( x, x,, x ) to deote a sample. R. I addto, we wrte Jot pmf (Dscrete Case): The jot pmf of (,,, ) s defed as f (x) = P ( = x, = x,, = x) for each ( x, x,, x) R. Hece for A R, P ( A) f(x) =. x A Jot pdf (Cotuous Case): The jot pdf of (,,, ) s the fucto that satsfes P ( A ) = f (x) d x = f ( x, x,, x ) dx dx dx. A A Jot cdf: The jot cdf of (,,, ) s defed as F(x) = P( x, x,, x ) for each ( x, x,, x) R. Expected Values: 3

4 Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Eg() g(x) f(x) dx or = Eg() g(x) f(x) = x R Margal dstrbuto of (,,, ): or f ( x, x,, x ) f( x,, x ) dx dx, = k k+ f ( x, x,, x k) = f k ( x (,, ),, x ) x. x R k+ Codtoal pdf or pmf of (,,, ): If f( x, x,, x ) > 0, the the codtoal pdf or pmf of (, k+, ) gve,, = x k = xk s f ( x, x x,, x ) f( x, x,, x )/ f( x,, x ). k+, k = k Defto 4.6.: Let m ad be postve tegers ad let p, p,, p be umbers satsfyg 0 p, =,, ad p = =. The the radom vector (,,, ) has a multomal dstrbuto wth m trals ad cell probabltes p, p,, p f the jot pmf s (,,, ) m! x x f x x x = p p x! x! o the set of ( x,, x ) such that each x s a oegatve teger ad x = m. = 4

5 Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Notes: m!. s kow as the multomal coeffcet whch deotes the umber of ways m objects ca be x! x! dvded to groups wth x the frst, x the secod,, x the th group.. Bomal s a specal case of the multomal dstrbuto where =. Theorem (Multomal Theorem): Let m ad be postve tegers. Let A be the set of vectors x = ( x,, x ) such that each x s a oegatve teger ad x = m. The for ay real umbers p,, p, ( ) = m! m x x p p p p + + = x A x! x!. Results:. The margal dstrbuto of s bomal( mp, ). p p. The codtoal dstrbuto of,, gve = x s multomal( m x;,, ). p p 3. The covarace of ad j s Cov(, j) = mpp j. Defto 4.6.5: Let,, be radom vectors wth jot pdf or pmf f (x,,x ). Let f (x ) deote the margal pdf or pmf of. The,, are called mutually depedet radom vectors, f for every (x,,x ), 5

6 Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 f (x,,x) f (x) f (x) f (x). = = = If the are all oe-dmesoal, the,, are called mutually depedet radom varables. Note: Parwse depedece does ot mply mutual depedece. Theorem 4.6.6: Let,, are mutually depedet radom varables. The E( g ( ) g ( )) = E( g ( )) E( g ( )). Theorem 4.6.7: Let,, are mutually depedet radom varables wth Z = + +. The M () t = M () t M () t. Z M (), t, M () t. Defe Example 4.6.8: Usg the mgf techque, show that f ~ gamma( α, β ), the Z = + + ~gamma( α + + α, β). Corollary 4.6.9: Let,, are mutually depedet radom varables wth ad b,, b fxed costats. Defe Z = ( a + b) + + ( a + b). The t b M () t = e M ( at) M ( a t). Z M (), t, M () t. Let a,, a 6

7 Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Corollary 4.6.0: Let,, be mutually depedet radom varables wth b,, b fxed costats. Defe Z = ( a + b) + + ( a + b). The. ~ ( ( ), ) µ + σ = = Z a b a ~ ( µ, σ ). Let a,, a ad Theorem 4.6.: Let,, be radom vectors. The Let,, are mutually depedet radom vectors f ad oly f there exst fuctos g(x )( =,, ), such that the jot pdf or pmf of Let,, ca be wrtte as f(x,,x ) = g (x ) g (x ). Theorem 4.6.: Let,, be radom vectors. Let g(x )( =,, ) be a fucto oly of x. The the radom varables U = g ( )( =,, ) are mutually depedet. Theorem: Let (,, ) are radom varables wth jot pdf f( x,, x ). Let A s the support set. Cosder the ew radom vector U = ( U,, U ), defed by U = g(). For a partto A0, A0,, A k of A : P( A 0) = 0 ad o each A, the trasformato s oe-oe from A to B, the we have x j = hj( u,, u)( j =,, ; =,, k). Let J be the Jacoba o A, the the jot pdf of ( U,, U ) s Example 4.6. ad k fu( u,, u ) = f ( h ( u,, u),, h( u,, u)) J. = 7

8 Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Defto: Let u = ( u,, u ) ad s the covarace matrx, whch σ j = σσ jρj. The multvarate ormal pdf s gve by f = Σ ( π ) Σ. T (x) exp (x-u) (x-u) / Propertes:. The margal dstrbuto s the multvarate ormal.. The correlato s ρ j. 3. Ay lear combato has the ormal dstrbuto. 4. The codtoal dstrbuto s the multvarate ormal. 5. They are mutually mpedet f ad oly f they are parwse depedet (parwse u-correlated). 6. If ay lear combato has the ormal dstrbuto, the the jot pdf s the multvarate ormal. 8