Probability & Random Variables

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Osborne Johnson
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1 Philadlphia Univrsit Dpartmnt o Commnication & Elctronics Enginring Probabilit & Random Variabls Chaptr or Instrctor Ibrahim N. Ab-Isbih absbih@philadlphia.d.jo Wbsit:

2 Ch.4: Mltipl Random Variabls. Vctor Random Variabls. Joint Distribtion and its Proprtis 3. Joint Dnsit and its Proprtis 4. Statistical Indpndnc 5. Distribtion and Dnsit o a Sm o Random Variabls 6. Cntral Limit Thorm Probabilit & Random Variabls - CH.4

3 : Vctor Random Variabls Lt and dnot two random variabls dind on a sampl spac S whr spciic vals o and ar dnotd b and. Thn an ordrd pair o nmbrs ma b considrd to b a random point in plan. Th point ma b takn a spciic val o a vctor random variabl. Th plan o all points in th rangs o and ma b considrd a nw sampl spac calld a joint sampl spac S J. Probabilit & Random Variabls - CH.4 3

4 As in th cas o on random variabl lt s din vnts A and B b A { } and B { } Th vnt A B dind on S corrsponds to th joint vnt { and } dind on S J. Probabilit & Random Variabls - CH.4 4

5 : Joint Distribtion and Its Proprtis Th probabilitis o th two vnts A={ } and B={ } hav distribtion nctions: W din th probabilit o th joint vnt { and } b a joint probabilit distribtion nction It shold b clar that and P P P P P A B Probabilit & Random Variabls - CH.4 5

6 Eampl: Assm that th joint sampl spac S J has onl thr possibl lmnts and 33. Th probabilitis o ths lmnts ar to b P=. P=.3 and P33=.5 Th distribtion nction: Probabilit & Random Variabls - CH.4 6

7 Joint Distribtion or Discrt Random Variabls: Th joint distribtion nction o discrt random variabls and is givn b: Eampl: N M P n m n m n m Probabilit & Random Variabls - CH.4 7

8 Probabilit & Random Variabls - CH.4 8 Proprtis o th Joint Distribtion:... 3 nction ing a nondcras is. 4 } { 5. P and. 6

9 Marginal Distribtion nctions: Proprt 6 abov stats that th marginal distribtion nctions obtaind b Eampl: and th marginal distribtion nctions: Probabilit & Random Variabls - CH.4 9

10 3: Joint Dnsit and Its Proprtis Th joint probabilit dnsit is dind b th scond drivativ o th joint distribtion nction: Probabilit & Random Variabls - CH.4 Joint Dnsit or Discrt Random Variabls: Th joint dnsit nction o discrt random variabls and is givn b: m N n M m n m n P

11 Eampl: Assm that th joint sampl spac S J has onl thr possibl lmnts and 33. Th probabilitis o ths lmnts ar to b P=. P=.3 and P33=.5 Th dnsit nction: Probabilit & Random Variabls - CH.4

12 Probabilit & Random Variabls - CH.4 Proprtis o th Joint Dnsit:. - - dd. d d. 3 d d d d 4. Marginal Distribtion nctions

13 Probabilit & Random Variabls - CH.4 3 } {. 5 dd P Proprtis o th Joint Dnsit: d d 6. Marginal Dnsit nctions

14 Marginal Dnsit nctions: Proprt 6 abov stats that th marginal distribtion nctions obtaind b d d and d d Eampl: th marginal dnsit nctions: Probabilit & Random Variabls - CH.4 4

15 Eampl: ind th val o b so that th ollowing nction is a valid joint dnsit nction g Soltion: b cos and all othr and thn b cos dd b b b d cos d Probabilit & Random Variabls - CH.4 5

16 Eampl: ind th marginal dnsit nctions whn th joint dnsit nction is givn b Soltion: Probabilit & Random Variabls - CH.4 6 d and d d

17 4: Statistical Indpndnc Th two random variabls and ar calld statisticall indpndnt i or Probabilit & Random Variabls - CH.4 7

18 Eampl: or prvios ampl Soltion: Thror th random variabls and ar not indpndnt. Probabilit & Random Variabls - CH.4 8 and

19 Eampl: Th joint dnsit o two random variabls and is Dtrmin i and ar indpndnt. Soltion: Thror th random variabls and ar indpndnt. Probabilit & Random Variabls - CH.4 9 /3 4 / 3 / 4 / /3 /3 4 / 4 / /3 4 / d d

20 5: Distribtion and Dnsit o a Sm o Random Variabls I W b a random variabl qal to th sm o two indpndnt random variabls and : W=+ thn th dnsit nction o W is th convoltion o thir dnsit nctions W w w d Probabilit & Random Variabls - CH.4

21 Eampl: ind th dnsit o W=+ whr Soltion: Probabilit & Random Variabls - CH.4 b a b b a a with d w w W

22 6: Cntral Limit Thorm Th cntral limit thorm sas that th probabilit distribtion nction o th sm o a larg nmbr o random variabls approachs a gassian distribtion Eampl: Probabilit & Random Variabls - CH.4 a a a a

Classical Linear Regression Model

Classical Linear Regression Model Classical Linar Rgrssion Modl Lt y t b th t-th random variabl, t, 2,,, whr is th sampl siz y Assumptions A: Linarity in th paramtr vctor b: E(y t ) Σ i ti β i : E(Y) Xb whr ti is th t-th obsrvation on