ENGI 4421 Joint Probability Distributions Page Joint Probability Distributions [Navidi sections 2.5 and 2.6; Devore sections

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1 ENGI 441 Jot Probablty Dstrbutos Page 7-01 Jot Probablty Dstrbutos [Navd sectos.5 ad.6; Devore sectos ] The jot probablty mass fucto of two dscrete radom quattes, s, P ad p x y x y The margal probablty mass fuctos are p ( x) p( x, y) p ( y) p( x, y) y x Example 7.01 Fd the margal p.m.f.s for the followg jot p.m.f. px, y y = 3 y = 4 y = 5 p x x = x = p y Lke ay other probablty mass fucto,, 1. a jot p.m.f. must be o-egatve for all (x, y) ad be coheret, px y x y It s easy to check that both codtos are satsfed example The radom quattes ad are depedet f ad oly f,, p x y p x p y x y A equvalet statemet s P x y P xp y x, y

2 ENGI 441 Jot Probablty Dstrbutos Page 7-0 I example 7.01, p 0 p , but Therefore ad are depedet, p x, 3 p x p 3 for x = 0 ad x = 1!]. [despte p 0, For ay two possble evets A ad B, codtoal probablty s defed by P[ A B] P[ A B], whch leads to the codtoal probablty mass fuctos P[ B] p( x, y) p( x, y) p ( y x) ad p ( x y). p ( x) p ( y) I example 7.01, px, y y = 3 y = 4 y = 5 p x x = x = p y

3 ENGI 441 Jot Probablty Dstrbutos Page 7-03 Jot Probablty Desty Fuctos [for bous questos oly] The cotuous aalogue to the dscrete jot probablty mass fucto s the jot probablty desty fucto, f (x, y). x, y dy dx 1. Jot desty fuctos are related to probablty statemets by tegrato over tervals: It must satsfy the codtos f x, y 0 x, y ad f P a b c d f x, ydx dy Margal p.d.f.s are defed by, ad, f x f x y dy Codtoal p.d.f.s are defed by f x, y f y x f f x ad x y d c b a f y f x y dx x, y y f f. Two cotuous dstrbutos are depedet f ad oly f f x, y f x f y x, y Example 7.0 (Navd, exercses.6, page 156, questo 0) [bous questo oly] Let deote the amout of shrkage ( %) udergoe by a radomly chose fbre of a certa type whe heated to a temperature of 10C. Let represet the addtoal shrkage ( %) whe the fbre s heated to 140C. The jot probablty desty fucto of ad s gve by 48xy 3 x 4 ad 0.5 y1 f x, y 49 0 otherwse (a) Fd P 3.5 ad 0.8. (b) Fd the margal probablty desty fuctos (c) Are ad depedet? Expla. f x ad f y (a) P 3.5 ad 0.8, f x y dx dy xy 48 dx dy x dx y dy x y

4 ENGI 441 Jot Probablty Dstrbutos Page 7-04 Example 7.0 (cotued) (b) 1 48x y 48 y 16 1 x f x f x, ydy dy x x (for 3 < x < 4 oly) ad x y 48 x 4 4y f y f x, ydx dx y y (for 0.5 < y < 1 oly). (c) For all (x, y) such that 3 < x < 4 ad 0.5 < y < 1, x 4y 48xy f x f y f x, y Therefore yes, ad are depedet. Wheever f x, y g x h y x, y o a rectagular doma alged wth the coordate axes (or o all of ), the radom quattes ad are depedet. These cocepts of jot probablty dstrbutos (both dscrete ad cotuous) ca be exteded to the cases of three or more radom quattes.

5 ENGI 441 Jot Probablty Dstrbutos Page 7-05 Expected Value or h x, y f, E[ h(, )] h( x, y) p( x, y) x y A measure of lear depedece s the covarace of ad : x y dx dy Cov[, ] E E[ ] E[ ] x y p( x, y) x y or, the cotuous case, Cov[, ] E E[ ] E[ ] x y f x, y dx dy Mapulate the double summato: Cov[, ] xy y x p( x, y) x y,,,, xy p x y y p x y x p x y p x y x y y x x y x y E E E Therefore Cov[, ] E E E for both dscrete ad cotuous radom quattes. Note that V[ ] = Cov[, ]. I Example 7.01, fd the covarace of ad : px, y y = 3 y = 4 y = 5 p x x = x = p y

6 ENGI 441 Jot Probablty Dstrbutos Page 7-06 Example 7.01 (cotued) E E E x y Cov, E E E Note that the covarace depeds o the uts of measuremet. If s re-scaled by a factor c ad by a factor k, the Cov c, k E c k E c E k ck E c E k E E E E Cov, ck ck A specal case s Vc Cov c, c c V. Ths depedece o the uts of measuremet of the radom quattes ca be elmated by dvdg the covarace by the product of the stadard devatos of the two radom quattes.

7 ENGI 441 Jot Probablty Dstrbutos Page 7-07 The correlato coeffcet of ad s, Corr, Cov[, ] E[ ] E[ ] E[ ] V[ ] V[ ] I Example 7.01, E V[ ] = E V[ ] = Example 7.0 For a jot uform probablty dstrbuto: (ad otg x x y y ): x y I geeral, for costats a, b, c, d, wth a ad c both postve or both egatve, Also: a b c d Corr, Corr,

8 ENGI 441 Jot Probablty Dstrbutos Page 7-08 Rule of thumb:.8 strog correlato.5. 8 moderate correlato.5 weak correlato I the example above, = 0.04 very weak correlato (almost ucorrelated)., are depedet p(x, y) = x y p x p y x p x y p y E E E Cov[, ] = It the follows that, are depedet, are ucorrelated ( = 0), but, are ucorrelated, are depedet. Couterexample (7.03): Let the pots show be equally lkely. The the value of s completely determed by the value of. The two radom quattes are thus hghly depedet. et they are ucorrelated!

9 ENGI 441 Jot Probablty Dstrbutos Page 7-09 Lear Combatos of Radom Quattes Let the radom quatty be a lear combato of radom quattes : a 1 the E E 1 a But t ca be show that V[ ] a a Cov[, ] 1 j1 j j depedet V[ ] V a 1 Specal case:, a1 1, a 1: E 1 ad V 1 Example 7.04 Two ruers tmes a race are depedet radom quattes T 1 ad T, wth , 4, Fd E T1 T ad V T1 T.

10 ENGI 441 Jot Probablty Dstrbutos Page 7-10 Dstrbuto of the Sample Mea If a radom sample of sze s take ad the observed values are 1,, 3,, the the are depedet ad detcally dstrbuted (d) (each wth populato mea ad populato varace ) ad two more radom quattes ca be defed: Sample total: T Sample mea: T E 1 Also V

11 ENGI 441 Pot Estmato Page 7-11 Ubased estmator A for Based estmator B for some ukow parameter : the ukow parameter : E[A] = E[B] Whch estmator should we choose to estmate? A mmum varace ubased estmator s deal. [See also Problem Set 7 Questo 8]

12 ENGI 441 Pot Estmato Page 7-1 Accuracy ad Precso (Example 7.05) [Navd secto 3.1; Devore secto 6.1] A archer fres several arrows at the same target. Error = Systematc error + Radom error (error) = (bas) + V[estmator] Estmator A for s cosstet ff E A V A 0 ad (as ) A partcular value a of a estmator A s a estmate.

13 ENGI 441 Pot Estmato Page 7-13 Sample Mea A radom sample of values 1,, 3,, s draw from a populato of mea ad stadard devato. The E, V ad the sample mea estmates. E, V But, f s ukow, the s ukow (usually). Sample Varace 1... S 1 ( 1) 1. 1 ad the sample stadard devato s S S 1 = umber of degrees of freedom for S. Justfcato for the dvsor ( 1) [ot examable]: Usg V E E for all radom quattes, E V E 1 E E 1 1 E E E E set set 1 V E V E 1.. d. 1

14 ENGI 441 Pot Estmato Page 7-14 S s the mmum varace ubased estmator of s the mmum varace ubased estmator of. Both estmators are also cosstet. ad Is a ull hypothess reject o Iferece Some Ital Cosderatos H o true (our default belef ), or do we have suffcet evdece to H o favour of the alteratve hypothess H A? H could be defedat s ot gulty or, etc. The correspodg The burde of proof s o H could be defedat s gulty or, etc. A H A. o o Bayesa aalyss: s treated as a radom quatty. Data are used to modfy pror belef about. Coclusos are draw usg both old ad ew formato. Classcal aalyss: Data are used to draw coclusos about, wthout usg ay pror formato. [Ed of Chapter 7]