Probability and Statistics

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Pearl Cameron
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1 Probability ad Statistics

2 Cotets. Multi-dimesioal Gaussia radom variable. Gaussia radom process 3. Wieer process

3 Why we eed to discuss Gaussia Process The most commo Accordig to the cetral limit theorem, may radom processes are Gaus sia radom process. e.g. resistace thermal oise, trasistor oise, atmospheric turbulece, cosmic oise. The easiest radom process(advatages of mathematics) Completely determied by the mea ad covariace Wide-sese statioary(wss) strictly-sese statioary(sss) After liear trasformatio, remais Gaussia radom process

4 . Oe-dimesioal Gaussia distributio ( x a) ~ (, ) ( ) exp{ } X N a f x Mea m E( X ) a X Variace DX ( ) X

5 .Oe-dimesioal Gaussia distributio Normalizatio (stadardizatio) Y X a ~ N(0,) y f( y) exp{ }

6 .Two-dimesioal Gaussia distributio ( X, X ) N a a,, r ~ (,, ) x a x a x a x a f x, x exp - r = (- ) r r X ~ N( a, ) X ~ N( a, ) r C CC Covariace cov( X, X ) r

7 .Two-dimesioal Gaussia distributio Normalizatio (stadardizatio) Y X a ~ N(0,) Y X a ~ N(0,) f y, y = exp - y - ryy y r (- r )

8 .Two-dimesioal Gaussia distributio x a x a x a x a f x, x = exp - - r + ( ) r -r Ucorrelatio is equivalet to mutual idepedece. x a x a f x, x exp - + x a x a exp - exp - f ( x ) f ( x)

9 .Two-dimesioal Gaussia distributio Ucorrelatio is equivalet to mutual idepedece.

10 .Two-dimesioal Gaussia distributio Matrix represetatio Let x x x a a a covariace matrix C C C r C C r correlatio coefficiet r C CC

11 .Two-dimesioal Gaussia distributio C r r C C r T r x a [ ] C r x a x - a C x - a x a x a -r x a x a x a x a - r

12 .Two-dimesioal Gaussia distributio x a x a x a x a f x, x exp - r + T x a x a x a x a C r x - a C x - a -r + -r = (- ) r r fx x x a C x a C T ( ) exp ( ) ( )

13 3.N-dimesioal Gaussia distributio N-dimesioal radom vector X X X X T Mea vector Covariace matrix a a a a T C C C C C C C C C C C cov( X, X ) r ij i j ij i j C cov( X, X ) ii i i i

14 3.N-dimesioal Gaussia distributio X ~ N( a, C) The probability desity of -dimesioal Gaussia distributio is fx x x a C x a C T ( ) exp ( ) ( ) C is positive defiite.

15 3.N-dimesioal Gaussia distributio As the probability desity fuctio, followig coditios: f x ( x) satisfies. fx( x) 0, xr. f ( ) d R x x x obviously Here is the proof:

16 3.N-dimesioal Gaussia distributio PROVE: f ( ) d R x x x C positive defiite matrix, L osigular matrix, so that C = LL T Let - y = L x -a, the x = Ly + a Jacobia determiat x L y C /

17 3.N-dimesioal Gaussia distributio so f ( ) d R x x x C T exp ( ) ( ) dxdx dx x a C x a C T / exp dydy dy yy C exp y y y dydy dy u / e du

18 3.N-dimesioal Gaussia distributio Ucorrelatio is equivalet to mutual idepedece. If they are ucorrelated, the C C / 0 0 i i - C / 0 / 0 / T xi a i exp ( x a) C ( x a) exp i i fx x x a C x a C x i a i exp / i f xi i i i T ( ) exp ( ) ( )

19 4.The margial distributio The margial distributio of multi-dimesioal Gaussia distributio remais Gaussia.

20 5. The statistical idepedece X, X,, X are the compoets of the -dimesioal Gaussia radom vectors, the ecessary ad sufficiet coditio of mutual statistical idepedece is pairwise ucorrelatio. X

21 6. Liear trasformatio Y L X L X L X Y L X L X L X Y L X L X L X Y = LX T X Y Y Y Y X X X L L L L L L L L L L T

22 6. Liear trasformatio fx x x a C x a C Let f T ( ) exp ( ) ( ) Whe a = 0 Γ L the X=ΓY ( y) f ( Γy) Y X J L C Γy J Γ / L y T exp ( Γy) C ( Γy)

23 6. Liear trasformatio T T exp y Γ C Γ L C y ad Let T T Γ C Γ L C L L C L LCL T T T F LCL, the F L C so that f Y ( ) exp F T y y F y

24 6. Liear trasformatio Whe the a 0, let b = La fy( y) exp y b F y b F T After liear trasformatio, multi-dimesioal Gaussia vectors remai multi-dimesioal Gaussia vectors.

25 Ⅱ. Gaussia radom process Defiitio process If ay fiite dimesioal distributio of the radom X ( t), t called a Gaussia radom process. is Gaussia distributio, the it is Gaussia process is a importat subclass of secod momet process. T

26 Ⅱ. Gaussia radom process The -dimesioal joit probability desity fuctio of Gaussia process X (t) fx x x a C x a C x T ( ) exp ( ) ( ) x t x t x t a C C C C C C C C C C a t a t a t T T, Cik E X ti ai X tk ak R t t a t a t X i k i k

27 Ⅱ. Gaussia radom process Gaussia process is secod momet process It s probability desity depeds oly o first ad secod momet. E[ X ( t)] Statioary Gaussia process If X ( t), t T is statioary, the,,, ;,,, =,,, ;,,, f x x x t t t f x x x t t t WSS SSS E[ X ( t)] SSS WSS Wide-sese statioary is equivalet to strictly-sese statioary.

28 Ⅱ. Gaussia radom process Coclusio Gaussia process is secod momet process. E[ X ( t)] Wide-sese Statioary is equivalet to strictly-sese statioary. Ucorrelatio is equivalet to mutual idepedece. Liear Trasformatio is Gaussia process.

29 Ⅱ. Gaussia radom process e.g. 5.- Set X(t) is a Gaussia radom process defied i [a, b], ad g t are two arbitrary ozero real fuctios, let b b Y g t X t dt a Y Y Y g t X t dt a Prove that ad are joitly Gaussia. gt

30 Ⅱ. Gaussia radom process e.g. 5.- Let Z(t)=Xcost + Ysit, If X ad Y are mutually idepedet Gaussia Radom variables, ad E[X]=E[Y]=0, D[X]=D[Y]=. Give: () The pdf of Z(t); () The pdf of R=sqrt(X^+Y^)

the first rigorous mathematical treatmet of the radom process describig Browia motio

31 III. Wieer process 88, Scottish botaist Robert Brow: flower polle exhibited persistet ad irregular radom motio 93, America mathematicia Norbert Wieer published the first rigorous mathematical treatmet of the radom process describig Browia motio 905, Germa physicist Albert Eistei published the first physically soud theory of the pheomeo

32 III. Wieer process Defiitio If a radom process has the followig properties:. PW [ (0) 0]. idepedet icremet process 3. Statioary icremets 4. the icremets followig Gaussia distributio

33 III. Wieer process Statistical average E[ W ( t ) W ( t )] 0 D[ W( t ) W( t )] t t E[ W ( t)] 0 D[ W ( t)] t R ( t, t ) E[ W( t ) W( t )] mi( t, t ) W Cotiuous i mea square

34 III. Wieer process N dimesioal pdf f ( x, x... x, t, t... t ) f ( x x ) f ( x x ) i 0 ( x x ) i i exp{ } ( t i ti ) ti ti ( ) ( ) -dimesioal pdf x f( x) exp{ } t ( ) t

35 III. Wieer process Wieer process, also kow as Browia motio process or Wieer Levi process. Wieer process Wt () No-statioary Gaussia process Idepedet icremet process, ad is homogeeous Distributio of icremet obeys Gaussia distributio Zero-mea, Wieer process ca be obtaied from ideal itegrator with statioary white Gaussia oise as iput.

36 HOMEWORK P397-6: Oly show the coclusio about X(t). Let Z(t)=Xcost+Ysit, where X ad Y are mutually idepedet Gaussia radom variables, ad E[X]=E[Y]=0, D[X]=D[Y]=. Please give: () Z(t) s probability desity fuctio; () The joit probability desity fuctio of Z(t) ad Z(t).

STATISTICAL METHODS FOR BUSINESS

STATISTICAL METHODS FOR BUSINESS UNIT 5. Joit aalysis ad limit theorems. 5.1.- -dimesio distributios. Margial ad coditioal distributios 5.2.- Sequeces of idepedet radom variables. Properties 5.3.- Sums