3. Several Random Variables

Transcription

1 . Several Random Variables. To Random Variables. Conditional Probabilit--Revisited. Statistical Independence.4 Correlation beteen Random Variables Standardied (or ero mean normalied) random variables.5 Densit Fnction o the Sm o To Random Variables. Probabilit Densit Fnction o a Fnction o To Random Variables.7 The Characteristic Fnction Concepts To Dimensional Random Variables Probabilit in To Dimensions Conditional Probabilit--Revisited Statistical Independence To Dimensional Statistics Correlation beteen Random Variables Densit Fnction o the Linear Combination o To Random Variables Mlti-inpt lectrical Circits Simlating Convoltion Integrals Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8

2 Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8 amples o Joint Densit Fnctions ercise -. To R.V. and have a oint pd o the orm and or k Find everthing. k Coeicient d d k d d d k d k k k k k Marginal densit nctions o and d 4 d Joint Distribtion o and dv d v F dv v F and or F

3 Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8 The conditional Probabilit that is greater than ½ given that =½. 4 4 d d F 4 d F F The conditional Probabilit that is less than or eqal to ½ given that =½. d d d F d F 8 4 F

4 ercise -. k ep or and There is no a nless the problem as spposed to be stated as k ep a or and bt then k and a are not necessaril compted separatel as and.! Overall the correct pd is Correlation ep or and ep ep or and ep or ep or d d ep d d a a ep epa d a ep ep ep ep ep d d ep ep ep ep Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 4 o 5 C 8

5 ercise -. Assme and independent Find Pr.5 ep or.5 ep or? That is the prodct o the random variables is positive. Pr Pr Pr Pr Pr Pr F F F F Find the distribtion or and based on the ranges deined or the absolte vale For or and or.5 ep F d or or.5 ep F d F F ep.5.5 ep F F F.5.5 ep ep.5.5 d.5.5 ep Pr Pr F F.5 ep. 89 Pr F F. Pr Pr F F. 8 F F F F Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 5 o 5 C 8

6 Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8 ercise -4. Again the tet is rong To R.V. have means o and variances o and 4 respectivel. Their correlation coeicient is.5. Find everthing or 7 Alternatel 7 4 Dierences 5 4 or or 4

7 Densit Fnction o the Sm o To Random Variables The smming (dierencing) o random variables creates a ne random variable. pect that the reslting probabilit distribtion and densit nction are dierent (ecept hen Gassians are involved). We ant to ind Pr F Pr For the eperiment let Figre in to dimensions ith the line =+. We kno the oint densit nction then an probabilit can be compted as v F d dv We kno that hich sas that and e can allo as the limitation in is incorporated in the irst ineqalit. Thereore F F d d I e assme that and are statisticall independent F : d d Taking the derivative to ind the densit nction reslts in a convoltion! d F df d d d d d d Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 7 o 5 C 8

8 Interpreting the math: this is the convoltion o the to densit nctions! The sm o random variables densit nction ma be ormed/derived as either o the to eqations (eqivalent convoltion reslts) Ths there are to eqivalent orms or the densit nction. d d d d Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 8 o 5 C 8

9 ample #: Uniorm densities in and Forming the densit in or and or d.5.5 d d =.5 Lines o =- -.5<<.5 =-.5 For For d d.5.5) d d.5.5) The convoltion o to rectangles makes a triangle. else Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 9 o 5 C 8

10 Application ample: This describes the error in the least signiicant bit o nmbers that are ronded to integers! Initiall a niorm distribtion beteen +/-.5. Ater smming nmbers that are ronded o the densit nction o the error changes the maimm possible error increases and the shape distribtes. Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8

11 ample #: Uniorm densities in and eponential densit in Forming the densit in or and ep or d Lines o =- d d = << << For For d ep d ep ep ep ep d ep d ep ep ep ep e Thereore e ep or ep or Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8

12 Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8 ample #: Independent Gassian densities in and ep and ep Forming the densit in d d d ep ep d ep To solve complete the sqare in and remember that the Gassian pd integrates to.. The remaining elements o the eqation (terms not in ) transer otside the integral. Once completed the reslt shold be: ep This is a Gassian that has a variance that is the sqare root o the sm o the variances o the individal Gassians (larger spreading ot bt still Gassian) and the mean is the sm o the individal means. This reslt (and the etension to mltiple random variables) is a maor reason h Gassian Noise models are so idel sed! This is also part o an empirical proo o the central limit theorem (Section -5). The MATALB eample demonstrates convolved eponential pds becoming Gassian-like.

13 %% % Gaseeian Convoltion % p. 4 Cooper and McGillam % clear close all; =:.:5; =ep(-); g=; igre; plot(); hold on; or ii = : g=.*conv(g); =.*(:length(g)-); plot(g); end label(''); label('g()') ais([ ]); hold o; igre plot(g); label(''); label('g()') g().5 g() stimation theor reminder ˆ ˆ Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8

14 HW -. A signal has a Raleigh densit nction and a mean vale o and is added to a noise N that is niorml distribted ith a mean vale o ero and a variance o. and N are statisticall independent and can be observed onl as =+N. a) Find sketch and label the conditional probabilit densit nction () as a nction o or = = and =. r r Raleigh: R r ep or r R R and R R R thereore and 4 ero mean niorm implies A Uniorm: n or A n A N A then N The densit nction in based on convoltion becomes r r dr A and A n n N R Bt since both and N are bonded the integration bonds and range o alloable vales mst be carell considered. N R dn N: -<n< Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 4 o 5 C 8

15 Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 5 o 5 C 8 Selecting dr r r R N or d or d ep ep or or ep ep or or ep ep ep Knoing the densit o e need to determine the conditional densit else or N Then N For = the alloed range o is <=< ep ep

16 Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8 ep ep For = the alloed range o is <=< ep ep ep ep ep For = the alloed range o is <=<8 8 ep ep ep 8 8 ep ep ep 8 8 ep ep ep

17 Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 7 o 5 C 8 b) I an observation ields a vale o = hat is the best estimate o the tre vale o? ˆ d ˆ 8 8 ep ep ep ˆ d 8 ep 8 ep ep ˆ d ep ep ep d d 8 8 ep 8 ep ep 8 Q Q d 8 ep ep 8 8 ep 8 ep ˆ Q Q and ep 4 ep ep 8 4 ep ˆ Q Q ˆ

18 Probabilit Densit Fnction o a Fnction o To Random Variables Fnctions o random variables create ne random variable. As beore epect that the reslting probabilit distribtion and densit nction are dierent. Assme that there is a nction that combines to random variables and that the nctions inverse eists. and the inverse and W W and W The original pd is ith the derived pd in the transorm space o g. Then it can be proven that: Pr W Pr or eqivalentl g d d d d mpiricall since the densit nction mst integrate to one or ininite bonds the transormed portion o one densit mst have the same volme as the original densit nction. Using an advanced calcls theorem to perorm a transormation o coordinates. g J g d d J d d Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 8 o 5 C 8

19 ample #4: Then and let W an arbitrar selection and describes the orard transormation and and describes the inverse transormation. The Jocobian is J Thereore g and integrating or all to ind g g d d As ma be epected the integral ma need to be solved sing nmerical methods. Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 9 o 5 C 8

20 ample #5: Compte the area here the nominal cm sqare IC die sie varies b independent niorm random variables alloing a +/-.5% tolerance. Then:.. or Derive the area densit nction ( ). Then and describes the orard transormation and and describes the inverse transormation and Thereore g g g d rect rect rect d.. rect d.. g or and Integrating to ind the distribtion or d ln 9.95 d ln.5 or or Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8

21 G ln d ln.5 d or or G ln 9.95 or 9.95 ln ln or To determine the area ithin.5% bonds Pr Pr G.5 G99.5 Pr Pr.5 ln ln ln Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8

22 The Characteristic Fnction Whenever o see convoltion and transormation epect to see eqations related to either the Forier Transorm or Laplace Transorm! The characteristic nction o a random variable is deined as: ep or ep d The inverse o the characteristic nction is then deined as: ep d Application: The densit nction o to smmed random variables here ep d and ep Since e alread kno d d d We can compte and solve or the densit nction as ep d Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8

23 Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 o 5 C 8 ample # Repeat eample # sing the characteristic nctions Uniorm densities in and eponential densit in or and or ep d ep ep ep d ep ep ep Then ep ep Solving or the densit nction d ep or d d d ep ep ep ep Find it or etra credit bt it shold reslt in or e or ep ep

24 Compting the irst moment sing the characteristic nction: Dierentiating the characteristic nction b : valating the nction at = d d d d ep d ep d ep d d Compting other moments is perormed similarl here: d n d n d n d n n ep d n n n n n d d Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 4 o 5 C 8

25 The Joint Characteristic Fnction Whenever o see convoltion and transormation epect to see eqations related to either the Forier Transorm or Laplace Transorm! The characteristic nction o a random variable and is deined as: v ep v or ep v d d The inverse o the characteristic nction is then deined as: ep v v d dv Note that v v v v v v Notes and igres are based on or taken rom materials in the corse tetbook: Probabilistic Methods o Signal and Sstem Analsis (rd ed.) b George R. Cooper and Clare D. McGillem; Oord Press 999. ISBN: B.J. Bain Spring 5 5 o 5 C 8