Chapter 8: MULTIPLE CONTINUOUS RANDOM VARIABLES

Transcription

1 Charles Boncelet Probabilit Statistics and Random Signals" Oord Uniersit Press 06. ISBN: Chapter 8: MULTIPLE CONTINUOUS RANDOM VARIABLES Sections 8. Joint Densities and Distribution unctions 8. Epected Values and Moments 8. Independence 8.4 Conditional Probabilities or Multiple Random Variables 8.5 Etended Eample: To Continuous Random Variables 8.6 Sums o Independent Random Variables 8.7 Random Sums 8.8 General Transormations and the Jacobian 8.9 Parameter Estimation or the Eponential Distribution 8.0 Comparison o Discrete and Continuous Distributions Summar Problems Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

2 8. Joint Densities and Distribution unctions Cumulatie Distribution unction:the probabilit o the eent that the obsered random ariable is less than or equal to the alloed alue and that the obsered random ariable is less than or equal to the alloed alue. Pr The deined unction can be discrete or continuous along the - and -ais. Constraints on the cumulatie distribution unction are:. 0 or and is non-decreasing as either or increases 5. and Analogies: a -dimensional probabilit moing rom scalars to ectors ( or more elements) Calc as compared to Calc &? Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

3 -D probabilit computations: P Think in terms o unions and intersections o -D boes in the plane igure.6-4 Point set associated ith the eent { }. igure.6-5 Point set or the eent { < < }. Then b inspection e can arrie at P Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

4 Joint Probabilit Densit unction (pd) The deriatie o the cumulatie distribution unction is the densit unction Properties o the pd include. 0 or and. d d Note: the olume o the -D densit unction is one.. u du d 4. d and d 5. Pr Etension to n dimensions Similar concepts etend this to 4 or n dimensions. d d The error associated ith sending a rocket into three dimensional space must be modeled and deined in at least three dimension mabe more Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 4 o 48 ECE 800

5 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 5 o 48 ECE 800 Uniorm Densit Eample The uniorm densit unction in to dimensions can be deined as: else and or 0 Determine the densit unction in (integrate or all ) d d or Similarl the densit unction in is or Note: this is a characteristic o random ariables that are independent and The inerse is also true i the joint pd and CD hae this propert then the random ariables and are independent!

6 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 6 o 48 ECE 800 Eercise -. Cooper and McGillem else and or A ep Determine A 00 ep d d A d d ep ep ep ep d A d d A ep ep 0 0 A A d A 6 A Determine the Distribution unction d d 00 ep 6 d d 0 0 ep ep 6 d 0 ep ep 6 ep ep ep ep 6 Then or 4 Pr 4 ep ep 4 4 Pr 4 ep ep 4 4 Pr

7 Epected Values and Moments The epected alues is computed as ma be epected All epected alues ma be computed using the Joint pd. There are correlation and coariance relationships to be included. Correlation and Coariance beteen Random Variables The deinition o correlation is gien as But most o the time e are not interested in products o mean alues but hat results hen the are remoed prior to the computation. Deeloping alues here the random ariable means hae been etracted is deined as computing the coariance This gies rise to another actor hen the random ariable means and ariances are used to normalize the actors or correlation/coariance computation. or eample the olloing deinition correlation coeicient is based on the normalized coariance Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 7 o 48 ECE 800

8 Reminder rom discrete multiple R.V. Letting The mean alue The ariance Thinking in terms o a product unction e deine Thereore either the coariance or correlation coeicient can be used. Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 8 o 48 ECE 800

9 8. Independence o multiple R.V. When the multiple ariables are independent there is signiicant simpliication in the aboe computations! or independent R.V. We ould epect and As ell as or and Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 9 o 48 ECE Independence can greatl simpli a multiple random ariable problem. As preiousl mentioned R.V. that are independent and haing identical probabilit distributions is a (IID) are oten assumed in order to greatl simpli modeling and simulation.

10 8.4 Conditional Probabilit (Again ith multiple R.V.) Multiple random ariables easil eist hen a single entit or een person is described in terms o multiple parameters. We ma epect the parameters to be related or the entit or person. In addition hen parameters are related knoing something about one can inluence the second parameter being considered. The tet oers the eample o a person s height and eight. The medical communit has oten published ideal height and eight charts or people to aspire to Note that the olloing is a dierent approach than our tetbook Using the Cumulatie Distribution unction (CD) deine Pr M Pr M Considering a smaller region in Leading to (or the interal going to zero engineers proo belo) and These are dierent rom the probabilit o a continuous distribution taking on a single alue in and 0 or 0 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 0 o 48 ECE 800

11 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800 An engineering deriation ollos to relate the pd: lim lim 0 0 du u Then taking the partial ith respect to The corresponding conditional densit unction is and similarl it can be shon that rom these equations it can also be seen that This proides a a to compute the joint densit unction based on a conditional densit unction.

12 Joint Densit to marginal densit computations. The joint densit total probabilit concepts can deine the and marginal densities. d and d Then rom the conditional densit relationship ith the joint densit We can replace the joint densit unctions in the total probabilit equations to deine the pd densities o and based on the conditional densities as d d or d d Baes Theorem To derie the multiple ariables Baes Theorem e return to Equating the right to elements result in. or Important Note in reie: the joint probabilit densit unction completel speciies: both marginal densit unctions and both conditional densit unctions. Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

13 Equialent ormula or discrete and continuous probabilit Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

14 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 4 o 48 ECE 800 Reieing Independent Random Variables Propert and Then and Independence simpliies required computations so pa attention to problems statements! Useul results o independence or joint densit unctions Thereore and Note: i the are independent knoing one does not help ith the other! It onl matters i and are correlated in some a and not independent.

15 Eample.6-9 Mapping to create a joint densit unction The eperiment inoles rolling one die haing equall likel outcomes rom to and Pr or i 6 We deine to ne random ariables Then the pms become pm pm pm i 6 or or or 456 or or or 456 or 4 or or 0 or or or 0 or 00 or 0 or 0 or 0 or or or 40 or 4 or 4 Note that: pm pm pm Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 5 o 48 ECE 800

16 Eample p. 6 Cooper and McGillem or independence. 6 5 or 0 and 0 Computing the marginal densities: d and 6 5 Note that 0 6 d Thereore the ariables are not independent! 0 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 6 o 48 ECE 800

17 Eercise -. rom Cooper and McGillem Assume and are independent ind Pr 0 Pr 0.5 ep or 0.5 ep or? That is the product o the random ariables is positie. 0 Pr 0 Pr 0 Pr 0 Pr 0 Pr ind the distribution or and based on the ranges deined or the absolute alue or or and or 0.5 ep d or or 0.5 ep d ep ep ep ep d ep ep Pr Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 7 o 48 ECE 800

18 8.5 Etended Eample A to dimensional problem Assume a triangular region has a pd alue ith magnitude scaling in. 0 0 Determining the alue o c But e can ork on the marginal densities irst or the alue o based on Determining the constant Thereore Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 8 o 48 ECE 800

19 We no kno 6 6 We can compute the marginal densit o use a similar concept to 6 6 Are and independent absolutel not! Computing means ariances etc Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 9 o 48 ECE

20 Conditional Probabilities or this particular problem And 6 6 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 0 o 48 ECE 800

21 And inall 4 0 A similar computation can be made or 4 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

22 Deining a ne R.V. based on and or the eample proided there can be a summed R.V. deined. (more coming) 0 0 Letting Z=+. 6 Deining the CD in terms o the joint densit unction in and 0 igure.-8 The region Cz (shaded) or computing the pd o Z +. Notes and igures are based on or taken rom materials in the course tetbook: Probabilit Statistics and Random Processes or Engineers 4th ed. Henr Stark and John W. Woods Pearson Education Inc. 0. Conceptuall this is one o the diicult pieces to get setting the equations & ariables! Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

23 Important note i and are sapped in the preious math ou get the same result! Knoing the CD the pd is 0 The mean nd moment and ariance are: Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

24 To check these alues and All the math orks!! Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 4 o 48 ECE 800

25 8.6 Sums o Independent Random Variables (Again) The general solution or the sum o independent random ariables can be approached as a conditional probabilit. (Note: not applicable in the preious problem not independent). Letting S=+. Using a concept rom total probabilit or the conditioned alue o Assuming and are actuall independent But the probabilit is a proper CD no orm the pd taking a deriatie in s Which in perorming a partial ith respect to s The pd o the independent R.V. sum is the conolution o the to pd unctions. Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 5 o 48 ECE 800

26 The sum o to IID eponential is the Erlang Densit unction λ λ 0 Letting S= +. Knoing this is a conolution it is important to think o the bounds o the integration. The sum must be positie or zero thereore 0 s. When the to eponentials oerlap the summation can begin at 0 but the summation bounds in the integral ariable can onl go rom 0 to s! Thereore and λ λ λ λ λ λ s λ λ 0 This can also be computed using the MG. Let to the tetbook. Sums and eighted sums are important problems in R.V. and occur oten! As an eample e oten use inite impulse response (IR) ilters. The are the eighted sum o consecutie discrete time samples! Each sample is assumed to contain IID random noise! No compute the aerage noise and aerage noise poer! Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 6 o 48 ECE 800

27 The Erlang Densit unction ; λ λ λ! 0 0 λ The mean alue can be shon to be λ rom the preious eample o summed IID eponentials and or k= λ λ λ The ariance can be shon to be λ rom the preious eample o summed IID eponentials and or k= λ λ λ As seen rom beore the Erlang Distribution/Densit unctions are related to the Eponential Distribution/Densit unctions. see: Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 7 o 48 ECE 800

28 Sums o IID R.V. S I N is knon the epected alue o the sum should be epected S E n But hat i the number o elements to sum is a R.V.? The second step is no longer alid! Using total probabilit e can deine another a or N an R.V. No S S S As k as knon or each sum and and N should be considered independent S and identiing the mean alue o the R.V. N The ariance becomes more un S S Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 8 o 48 ECE 800

29 As k as knon or each sum and and N should be considered independent Considering S k k k No continuing And k k S k k The ariance becomes S k k k and k S Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 9 o 48 ECE 800

30 Stark and Woods eample o the sum o to R.V. Eample.-4 Z=+ here is eponential and is uniorm. ep or 0 rect We are conoling a rectangular indo ith an eponential. irst pick the bounds on the conolution that ill be most conenient (keep the uniorm distribution in place and moe the eponential). Then there are three regions o integration to consider: ) the densit unctions not oerlapping (z<-) ) the densit unctions starting to oerlap (-<z<) ) the densit unctions completel oerlapping (<z) Computations using: igure.-0 Relatie positions (z ) and () or (a) z < ; (b) z < ; (c) z >. Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 0 o 48 ECE 800

31 In region In region 0 0 ep ep ep ep ep ep In region ep ep ep ep ep ep ep epep The result can be stated as 0 ep epep `The result then appears in ig..- igure.- The pd Z (z) rom Eample.-4. Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

32 8.8 General Transormations and the Jacobian The tetbook solution sequence states:. Let =g() be the transormation. It is desired to compute the densit o.. Rather than computing the densit unction directl it ma be better to compute (or consider the computation o) the CD unction:. No comes the crucial step: inert the transormation to get a probabilit on and not g(). 4. Write the probabilit in terms o the distribution unction o as a unction o. 5. Dierentiate ith respect to to get the densit unction o. The simple eamples inole linear one to ong mappings. Linear mapping The CD o Creating the probabilit in Epress this in terms o the CD o Continuing on the net page dierentiating or the pd Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

33 Dierentiating or the pd 0 0 or This is diagramed in the tetbook as Introducing the Jacobian or the linear sstem transormation the /a is the Jacobian o the transormation. As described in notes rom the preious chapter hen looking at the summation e are interested in ; Where J(;) represents the Jacobian a partial deriatie unction o the ariables deined. and Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 o 48 ECE 800

34 Appling this technique to the linear transormation Pr Pr 0 or a negatie the integration bounds change Pr 0 Pr 0 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 4 o 48 ECE 800

35 Probabilit Densit unction o a unction o To Random Variables Notes and igures are based on or taken rom materials in the course tetbook: Probabilit Statistics and Random Processes or Engineers 4th ed. Henr Stark and John W. Woods Pearson Education Inc. 0. unctions o random ariables create ne random ariable. As beore epect that the resulting probabilit distribution and densit unction are dierent. Assume that there is a unction that combines to random ariables and that the unctions inerse eists. Z and W and the inerse Z W and Z W The original pd is ith the deried pd in the transorm space o z Then it can be proen that: Pr z Z z W Pr g. or equialentl z z g z dz d d d Empiricall since the densit unction must integrate to one or ininite bounds the transormed portion o one densit must hae the same olume as the original densit unction. Using an adanced calculus theorem to perorm a transormation o coordinates. [ Using the determinant o the Jacobian) g z z z z z z And the integrals become. z J z z g z dz d z z z z J dz d Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 5 o 48 ECE 800

36 Transormations or Multidimensional Densities p.. The tetbook generalizes the Jacobian to and n dimensional to n dimensional translation! The Jacobian shon is an nn matri that needs to hae a determinant taken to determine the multiplication actor! The original random ariables are deined as (as a ector o R.V.) The ne random ariables are also deined (as a ector R.V.) here There is an assumption o an inertable unction so that Using the determinant o the Jacobian The Transormed Densit unction becomes ; Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 6 o 48 ECE 800

37 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 7 o 48 ECE 800 or a linear problems that ma seem amiliar Notes and igures are based on or taken rom materials in the course tetbook: Probabilit Statistics and Random Processes or Engineers 4th ed. Henr Stark and John W. Woods Pearson Education Inc. 0. Gien: We can generate V and W Thereore W V and W V orming the Jacobian and taking the determinant J and det J inall e arrie at W V OK but hat about just V or W V= + d W V V d V Use a change in ariable: z and d dz dz z z V or and independent dz z z V Conolution W=- d W V W d W Use a change in ariable: z and d dz dz z z W or and independent dz z z W Correlation

38 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 8 o 48 ECE 800 Eample.4- Stark and Woods Gien: ep We can generate V 5 and W W V 5 inding the inerse W V 5 W V 5 5 Thereore W V 5 and W V 5 J and 5 det J Thereore W V 5 Gien the initial joint densit unction 5 ep W V ep W V ep W V

39 Eample 8.7 on p. S and In ector and matri notation the orard transormation becomes The Inerse Transormation Becomes. Thus inerse matri is the Jacobian S and 0 0 J 0 Thereore 0 or and IID and λ λ 0 We hae Continuing 0 λ λ λ λ λ λ Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 9 o 48 ECE 800

40 Determining the marginal densit in S the integral in can onl appl rom 0 to s and not 0 to ininit λ λ or the conditional probabilit desired in the problem λ λ s λ λ λ λ s λ λ Thereore gien a alue o S=s the alue o can be described as a uniorm random ariable rom 0 to s. This makes sense as + = S hich necessaril requires to be greater than or equal to zero and less than s. s Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring o 48 ECE 800

41 Eample 8.6 on p. R and Θtan The Inerse Transormation becomes. R cosθ and R sinθ Thus Jacobian matri J cosθ sinθ r sinθ r cosθ The determinant is J r cos Θ r sin Θ Thereore Θ r cosθr sinθ J r cosθr sinθ I the original joint R.V. unctions as uniorml distributed in a unit circle region here Continuing on the net page Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 4 o 48 ECE 800

42 Then Θ r cosθr sinθ J r 0Θ π Θ 00Θ π Do determine the marginal densit unctions: Θ Θ Θ Θ 0 π 0 and Θ Θ r Θ Θ r r r Θ 0 Θ 0 Θ π Θ 0 Θ π Note that Θ Θ 00Θ π The ne random ariables are independent! Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 4 o 48 ECE 800

43 Simpliing a preious problem: Eample.- Boncelet HW problem 8.0!: Eample #4 in Cooper and McGillem: V and let W an arbitrar selection Then and describes the orard transormation and and describes the inerse transormation. The determinant o the Jocobian is J 0 z Thereore z z z z z V W z V W Then integrating or all to ind (remember that d V V d V V W d V V and d Note: does the sapping o elements in the joint distribution reall matter or V? J Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring 08 4 o 48 ECE 800

44 Boncelet HW 8.0 I and are IID U(0 ) let Z =. a) What are the mean and ariance o Z? (Hint: ou can anser this ithout knoing the densit o Z.) b) (hard) What are the densit and distribution unction o Z? c) (hard) Calculate the mean and ariance o Z b integrating against the densit? Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring o 48 ECE 800

45 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring o 48 ECE 800

46 .5 Additional Eamples Stark and Woods p. 00 Notes and igures are based on or taken rom materials in the course tetbook: Probabilit Statistics and Random Processes or Engineers 4th ed. Henr Stark and John W. Woods Pearson Education Inc. 0. I there is interest see me or eample solutions to these particular problems. Eample.5-: rectangular to polar coordinate transormation Eample.5-: rectangular to magnitude and diision Eample.5-: rectangular coordinate angular rotation b theta Eample.5-4: using ZW to perorm to magnitude Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring o 48 ECE 800

47 Uniorm Densit Eample The uniorm densit unction in to dimensions can be deined as: or and 0 else Determine the densit in d d or Similarl or Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring o 48 ECE 800

48 Notes and igures are based on or taken rom materials in the course tetbook: Charles Boncelet Probabilit Statistics and Random Signals Oord Uniersit Press ebruar 06. B.J. Bazuin Spring o 48 ECE 800 Distribution d d d d Detailed distribution in the entire -plane and or and or and or and or and or 0