Probability, Random Variables, and Processes. Probability

Transcription

1 Prbability, Randm Variables, and Prcesses Prbability Prbability Prbability thery: branch f mathematics fr descriptin and mdelling f randm events Mdern prbability thery - the aximatic definitin f prbability - intrduced by Klmgrv in [Klmgrv, 1933] Octber 25, / 18

2 Prbability, Randm Variables, and Prcesses Prbability Definitin f Prbability Experiment with an uncertain utcme: randm experiment Unin f all pssible utcmes ζ f the randm experiment: certain event r sample space f the randm experiment - O Event: subset A O Prbability: measure P (A) assigned t A satisfying the fllwing three axims 1 Prbabilities are nn-negative real numbers: P (A) 0, A O 2 Prbability f the certain event: P (O) = 1 3 If {A i : i = 0, 1, } is a cuntable set f events such that A i A j = fr i j, then ( ) P A i = P (A i ) (1) i i Octber 25, / 18

3 Prbability, Randm Variables, and Prcesses Prbability Independence and Cnditinal Prbability Tw events A i and A j are independent if P (A i A j ) = P (A i ) P (A j ) (2) The cnditinal prbability f an event A i given anther event A j, with P (A j ) > 0 is P (A i A j ) = P (A i A j ) P (A j ) (3) With direct cnsequence: Bayes therem P (A i A j ) = P (A j A i ) P (A i) P (A j ) with P (A i ), P (A j ) > 0 (4) Definitins (2) and (3) als imply that, if A i and A j are independent and P (A j ) > 0, then P (A i A j ) = P (A i ) (5) Octber 25, / 18

4 Prbability, Randm Variables, and Prcesses Randm Variables Randm Variables Randm variable S: functin f the sample space O that assigns a real value S(ζ) t each utcme ζ O f a randm experiment Cumulative distributin functin (cdf) f a randm variable S: F S (s) = P (S s) = P ( {ζ : S(ζ) s} ) (6) Prperties f cdf: nn-decreasing, F S ( ) = 0, and F S ( ) = 1 N-dimensinal cdf, jint cdf, r jint distributin: F S (s) = P (S s) = P (S 0 s 0,, S N 1 s N 1 ) (7) with S = {S 0,, S N 1 } T being a randm vectr Jint cdf f tw randm vectrs X and Y F XY (x, y) = P (X x, Y y) (8) Octber 25, / 18

5 Prbability, Randm Variables, and Prcesses Randm Variables Cnditinal Cdfs Cnditinal cdf f randm variable S given event B with P (B) > 0 F S B (s B) = P (S s B) = P ({S s} B) P (B) (9) Cnditinal cdf f a randm variable X given anther randm variable Y F X Y (x y) = F XY (x, y) F Y (y) = P (X x, Y y) P (Y y) (10) Cnditinal cdf f a randm vectr X given anther randm vectr Y is given by F X Y (x y) = F XY (x, y)/f Y (y) Octber 25, / 18

6 Prbability, Randm Variables, and Prcesses Randm Variables Cntinuus Randm Variables If cdf F S (s) is a cntinuus functin: prbability density functin (pdf) f S (s) = df S(s) ds Prperties f pdfs: f S (s) 0, Unifrm pdf: F S (s) = s f S(t) dt = 1 f S (t) dt (11) Laplacian pdf: Gaussian pdf: f S (s) = 1/A fr A/2 s A/2 A > 0 (12) f S (s) = 1 e s µ S 2/σS σ S > 0 (13) σ S 2 f S (s) = 1 e (s µ S) 2 /(2σ 2 S ) σ S > 0 (14) σ S 2π Octber 25, / 18

7 Prbability, Randm Variables, and Prcesses Randm Variables Generalized Gaussian Distributin f S (s) = β 2αΓ(1/β) β e ( x µ /α) Γ(z) = 0 e t t z 1 dt (15) Octber 25, / 18

8 Prbability, Randm Variables, and Prcesses Randm Variables Jint and Cnditinal pdfs N-dimensinal pdf, jint pdf, r jint density f S (s) = N F S (s) s 0 s N 1 (16) Cnditinal pdf r cnditinal density f S B (s B) f a randm variable S given an event B f S B (s B) = df S B (s B)/ds (17) Cnditinal density f a randm vectr X given anther randm vectr Y f X Y (x y) = f XY (x, y) f Y (y) (18) Octber 25, / 18

9 Prbability, Randm Variables, and Prcesses Randm Variables Discrete Randm Variables Discrete randm variable S: if its cdf F S (s) represents a staircase functin S takes values f cuntable set A = {a 0, a 1,...} Prbability mass functin (pmf): Cdf f discrete randm variable p S (a) = P (S = a) = P ( {ζ : S(ζ)= a} ) (19) F S (s) = a s p(a) (20) Binary pmf: A = {a 0, a 1 } p S (a 0 ) = p, p S (a 1 ) = 1 p (21) Unifrm pmf: A = {a 0, a 1,, a M 1 } p S (a i ) = 1/M a i A (22) Gemetric pmf: A = {a 0, a 1, } p S (a i ) = (1 p) p i a i A (23) Octber 25, / 18

10 Prbability, Randm Variables, and Prcesses Randm Variables Jint and Cnditinal pmfs N-dimensinal pmf r jint pmf fr a randm vectr S = (S 0,, S N 1 ) T p S (a) = P (S = a) = P (S 0 = a 0,, S N 1 = a N 1 ) (24) Jint pmf f tw randm vectrs X and Y : p XY (a x, a y ) Cnditinal pmf p S B (a B) f a randm variable S given an event B, with P (B) > 0 p S B (a B) = P (S = a B) (25) Cnditinal pmf f a randm vectr X given anther randm vectr Y p X Y (a x a y ) = p XY (a x, a y ) p Y (a y ) (26) Octber 25, / 18

11 Prbability, Randm Variables, and Prcesses Randm Variables Example fr Jint pmf Fr example, samples in picture and vide signals typically shw strng statistical dependencies Belw: histgram f tw hrizntally adjacent sampels fr the picture Lena Relative frequency f ccurence Amplitude f adjacent pixel Amplitude f current pixel Octber 25, / 18

12 Prbability, Randm Variables, and Prcesses Randm Variables Expectatin Expectatin values r expected values f cntinuus randm variables S E {g(s)} = f discrete randm variables S g(s) f S (s) ds (27) E {g(s)} = a A g(a) p S (a) (28) Imprtant expectatin values are mean µ S and variance σs 2 µ S = E {S} and σs 2 = E { (S µ s ) 2} (29) Expectatin value f a functin g(s) f a set N randm variables S = {S 0,, S N 1 } E {g(s)} = g(s) f S (s) ds (30) R N Octber 25, / 18

13 Prbability, Randm Variables, and Prcesses Randm Variables Cnditinal Expectatin Cnditinal expectatin value f functin g(s) given an event B, with P (B) > 0 E {g(s) B} = g(s) f S B (s B) ds (31) Cnditinal expectatin value f functin g(x) given a particular value y fr anther randm variable Y E {g(x) y} = E {g(x) Y =y} = g(x) f X Y (x, y) dx (32) Octber 25, / 18

14 Prbability, Randm Variables, and Prcesses Randm Prcesses Randm Prcesses Series f randm experiments at time instants t n, with n = 0, 1, 2,... Outcme f experiment: randm variable S n = S(t n ) Discrete-time randm prcess: series f randm variables S = {S n } Statistical prperties f discrete-time randm prcess S: N-th rder jint cdf F Sk (s) = P (S (N) k s) = P (S k s 0,, S k+n 1 s N 1 ) (33) Cntinuus randm prcess f Sk (s) = N s 0 s N 1 F Sk (s) (34) Discrete randm prcess F Sk (s) = a A N p Sk (a) (35) A N prduct space f the alphabets A n and p Sk (a) = P (S k = a 0,, S k+n 1 = a N 1 ) (36) Octber 25, / 18

15 Prbability, Randm Variables, and Prcesses Randm Prcesses Statinary Randm Prcess Statinary randm prcess: statistical prperties invariant t a shift in time N-th rder jint cdf F Sk (s), pdf f Sk (s), and pmf p Sk (a) are independent f time instant t k and are dented by F S (s), f S (s), and p S (a), respectively N-th rder autcvariance matrix C N = E { (S (N) µ N )(S (N) µ N ) T } (37) is a symmetric Teplitz matrix 1 ρ 1 ρ 2 ρ N 1 ρ 1 1 ρ 1 ρ N 2 C N = σs 2 ρ 2 ρ 1 1 ρ N ρ N 1 ρ N 2 ρ N 3 1 (38) Fr Teplitz matrices, see the standard reference [Grenander and Szegö, 1958] and the tutrial [Gray, 2005] Octber 25, / 18

16 Prbability, Randm Variables, and Prcesses Randm Prcesses Memryless and i.i.d. Randm Prcesses Memryless randm prcess: randm prcess S = {S n } fr which the randm variables S n are independent Independent and identical distributed (iid) randm prcess: statinary and memryless randm prcess N-th rder cdf F S (s), pdf f S (s), and pmf p S (a) fr iid prcesses, with s = (s 0,, s N 1 ) T and a = (a 0,, a N 1 ) T F S (s) = N 1 k=0 F S (s k ), f S (s) = N 1 k=0 f S (s k ), p S (a) = N 1 k=0 p S (a k ) (39) F S (s), f S (s), and p S (a) are the marginal cdf, pdf, and pmf, respectively Octber 25, / 18

17 Prbability, Randm Variables, and Prcesses Randm Prcesses Markv Prcesses Markv prcess: future utcmes d nt depend n past utcmes, but nly n the present utcme, Discrete prcesses P (S n s n S n 1 =s n 1, ) = P (S n s n S n 1 =s n 1 ) (40) p Sn (a n a n 1, ) = p Sn (a n a n 1 ) (41) Example fr a discrete Markv prcess (calculate p(a)) a a 0 a 1 a 2 p(a a 0) p(a a 1) p(a a 2) p(a) Octber 25, / 18

18 Prbability, Randm Variables, and Prcesses Randm Prcesses Markv Prcesses II Cntinuus Markv prcesses f Sn (s n s n 1, ) = f Sn (s n s n 1 ) (42) Given zer-mean iid prcess Z = {Z n }, cntinuus Markv prcess S = {S n } with mean µ S Variance σ 2 S S n = Z n + ρ (S n 1 µ S ) + µ S, with ρ < 1 (43) f statinary Markv prcess S σ 2 S = E { (S n µ S ) 2} = E { (Z n ρ (S n 1 µ S ) ) 2} = Gauss-Markv Prcess, ρ = 0.9, µ S = σ2 Z 1 ρ 2 (44) s(t) t Octber 25, / 18