Statistical Signal Processing

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Cornelius Wheeler
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1 ELEG-66 Statistical Sigal Processig Pro. Barer 6 Evas Hall barer@udel.edu

2 Goal: Give a discrete time sequece {, how we develop Statistical ad spectral represetatios Filterig, predictio, ad sstem idetiicatio algorithms Optimizatio methods - Statistical - Adaptive

3 a methods assume that { is determiistic. Real world sigals are usuall statistical i ature. Thus, -,,, ca be iterpreted as a sequece o radom variables. - We begi b aalzig each observatio as a R.V. - The, to capture depedecies, we cosider radom vectors,,, N-, ~ N,

4 Radom Variables For a space S, the subsets, or evets o S have associated probabilities. To ever evet δ, we assig a umber δ, which is called a R.V. The distributio uctio o is F < < Properties: F, F F is cotiuous rom the right F F < F F

5 Eample: air toss o two cois Evetsδ Prob. Xδ Yδ HH ¼ - HT ¼ - TH ¼ - TT ¼ 4 5 This ields dieret distributio uctios F HH, HT / F HH, HT, TH / 4 F F

6 The probabilit desit uctio is deied as, df d or Thus F F d d Tpes o distributios: Cotiuous: Discrete: i i F F i P i I which case Pδ i i i ied: discotiuous but ot discrete. 5

7 6 Distributio eamples Uiorm: b a b a U <, ~ else b a a b ], [ Gaussia:, ~ σ µ N σ µ πσ e b a a b a b F µ µ F /

8 7 Biomial: Eample: Toss a coi times. What is the probabilit o gettig heads? For p q < m m q p F q p q p m δ For 9, pq/, Probabilit o tail Probabilit o head F

9 Coditioal Distributios The coditioal distributio o give evet has occurred is F, Eample: Suppose { a The F, a a I a, what happes? 8

10 I this case ad, a a F a a I a, the F a F F a Suppose F What does F loo lie? a 9

11 F a F As beore F F F F ad F d F a F

12 Eample: Toss a air coi our times, let be the umber o heads. Recall p q I this case / 6 / 4 / 8

13 F Suppose at least oe lip produces a head No heads 5 6 6

14 What is,, / / 6 5 / / \

15 Total Probabilit ad Baes Theorem Let,,, orms a partitio o S, i.e. U i i S ad i I i j j φ The F F i Pr i i i Pr i i A B A, B B B A Pr B A 4

16 5 From this we get Pr F F ad Pr b itegratio d Baes Theorem: Pr Pr Pr { d

17 Fuctios o a R.V. Let ad g be RVs such that g The F g R Where R { : g I g What is R? 6

18 F Pr Pr F F Eample: Let ~ N µ, σ Ad i > U i F 7

19 8 To determie the desit o g i terms o, loo at g Pr Pr Pr Pr d d d d d d d d

20 9 Note that / g d d d d d d d d Similarl g d d ad g d d Thus d g d g d g d or g g g

21 I geeral, or g, let,, be the roots g g the g g Eample: suppose ~ U,... ad - -

22 g case : ± g g / / / case : 4 g 6 / /.5 4 5

23 Eample: let ~ N µ, σ ad e The g ad g e Also, there is ol oe solutio l Thereore g e or l e l l

24 or l > or e e σ µ σ µ σ π πσ Log ormal desit

25 Distributio o F For a RV with cotiuous distributio F, the RV F is uiorm o [,]. Proo: Note << Sice g F g ad g F 4

26 Thus the uctio g F perorms the mappig U uiorm [,] F pd Similarl U F Uiorm pd pd Sthesis: U F F F 5

27 ea ad variace Coditioal mea E { d E { d Eample: I { a The E { d a a d d 6

28 For a uctio o a RV, g, E { Eample: Suppose d g g d The E { g g d F d 7

29 The variace is deied as σ d Where E { Thus, σ E { E { E { Eample: ~ N, σ σ e πσ is smmetric about E{ also d e σ d πσ 8

30 9 rearragig b dieretiatig σ πσ π σ πσ σ σ σ d e d e d e or { σ E

31 omets omets d E m { cetral momets d E { µ From the biomial theorem m E E { { µ µ σ µ µ µ m m

32 Eample: ~ N, σ The E { σ Proo: For odd { E d Eve uctio Odd uctio To prove the secod part, use the act that e α d π α

33 Dieretiate both sides with respect to α, times e α d let α σ, the π α e σ d σ π Lettig ad rearragig π σ e σ d σ

34 Variace is a measure o a RV s cocetratio aroud its mea Tchebche Iequalit For a ε>, Pr ε σ ε Proo: Pr ε ε d ε d ε d -ε ε

35 4 But ε σ d d ad sice ε ε ε ε ε σ ε ε d d ε σ ε

36 5

Stochastic Processes

Stochastic Processes Review o Elemetar Probabilit Lecture I Hamid R. Rabiee Fall 20 Ali Jalali Outlie Histor/Philosoph Radom Variables Desit/Distributio Fuctios Joit/Coditioal Distributios Correlatio Importat