??430 ??f?a?e? ?p????????e? ?a??

1
??430 ??f?a?e? ?p????????e??a??µa 2

• ?e?????? Te???a? ???a??t?t??

2
Sp??da??t?ta t?? st??ast???? d?ad??as???

• ?? t??a?e? d?ad??as?e? ?a? µetaß??te? µa?
  ep?t?ep??? ?a ?e?????µaste p?s?t?te? ?a? s?µata
  p?? de? ta ?e???µe e? t?? p??te???
• ?a ded?µe?a ?a? ta s?µata p?? µetad?d??ta? µesa
  ap? ta t??ep????????a?a s?st?µata ?e?????ta?
  t??a?a.
• ? ????ß??, ?? pa?eµß??e?, ?? pa?aµ??f?se?? ?a? ??
  d?a?e??e?? (fading) p?? e?sa???ta? ap? t? ?a?a??
  ep?s?? p??s?µ??????ta? µe st??ast??e?
  d?ad??as?e?.
• ???µa ?a? t? ???t???? a???p?st?a? µetad?s?? (BER-
  Bit Error Rate ? p??a??t?ta sfa?µat?? bit)
  e?f?a?eta? µe p??a??-?e???t????? ?????

3
???a?a ?e????ta

• ?ta? e?te???µe e?a t??a?? pe??aµa, µp????µe ?a
  ???s?µ?p???-s??µe s?µß??a t?? ?e???a? s?????? ??a
  ?a pe????a???µe ta d??ata ap?te?esµata.
• ?a?ade??µa ??????µe e?a ?a??.
• ???ata ap?te?esµata S 1,2,3,4,5,6
• Ge????? e??a? ?a?e ?p?s????? d??at??
  ap?te?esµat?? ?1,2
• S?µp????µat??? ?e????? t?? ? e??a? t? S A
  3,4,5,6
• ?? s????? ???? t?? ap?te?esµat?? e??a? t? s??????
  ?e????? S
• ? ? ????? ap?te?esµ?t?? ? ? ????? de??µat??
• ?? ?e?? ?e????? e??a? t? ?
• ? µetad?s? e??? bit, p.?., e??a? e?a t??a??
  pe??aµa

4
???a??t?ta

• ? p??a??t?ta P(A) e??a? e?a? a???µ?? ? ?p????
  µet?a t?? p??a??fa?e?a t?? ?e????t?? ?.
• St?? ???? de??µat?? S1,2,3,4,5,6 t??
  p??????µe??? pa?ade??µat?? a? ?1,2 t?te ?(?)
  1/3 ?a?
• ?(a?t?? ap?te?esµa)1/2
• ????µata t?? ?e???a? p??a??t?t??
• ??de? ?e????? e?e? a???t??? p??a??t?ta P(A)?0
• P(A)?1 ?a? P(A) 1 ? A S.
• ?? ? ?a? ? e??a? d?? ?e?a ?e????ta d??. a?
  ????
• t?te P(A ? B) P(A) P(B).
• O?e? ?? a??e? ?d??t?te? t?? p??a??t?t?? e??a?
  ap?????a a?t?? t?? a???µat??

5
??a??aµµata Venn
6
S?ese?? µeta?? t??a??? ?e????t??

• ? ap? ?????? p??a??t?ta t?? ? ?a? ? e??a? ?
  p??a??t?ta ?a s?µß??? ?a? ta d?? ?e????ta
  P(A,B) P(A ? B)
• ?p? s?????? p??a??t?ta P(AB) P(A,B) / P(B)
• ???a? ? p??a??t?ta ?t? ?a s?µße?
  t? ? ded?µe??? ?t? s???ß? t? ?
• ?ts? P(A,B) P(A)P(B)
  P(??)P(?)
• Stat?st??? a?e?a?t?s?a
• ?a ?e????ta ? ?a? ? e??a? stat?st??a a?e?a?t?ta
  a?
• P(A,B) P(A) P(B)
• ?? ta ? ?a? ? e??a? a?e?a?t?ta t?te
• P(AB) P(A) ?a? P(BA) P(B)
• ?a?ade??µa ?a ap?te?esµata t?? ????? d?? ?a????
  ? ta ap?te?esµata t?? ????? t?? ?d??? ?a????
  d?? f??e? (e?t?? a? e??a? pe??a?µe??)

7
?a?ade??µa stat?st???? e?a?t?s??

• ?st? S o ????? ap?te?esµat?? t?? pe??aµat?? ?????
  t?? ?a????. Te??e?ste ta ?e????ta ?3 ?a?
  ?1,2,3,6
• ?(?)1/3 ?a? ?(?)4/62/3
• ?(??) 1/4 ?(?,?)/?(?) ?(?,?)/(2/3) ?
• ? ?(?,?) (1/4)(2/3) 1/6
• ?p?te ?(?)?(?) (1/3)(2/3) 2/9 ? 1/6
  ?(?,?)
• ???ad? ta ?e????ta ? ?a? ? e??a? e?a?t?µe?a
• ??s? e??a? ? ?(??) ??
• ?? e?a?t?s? e???? ta ?e????ta G4 ?a? ? ?

8
1? ?a?ade??µa stat?st???? a?e?a?t?s?a?

• Te??e?ste t?? ???? O p?? ap?te?e?ta? ap? ta 52
  ap?te?esµata t?? t??a??? pe??aµat?? p?? e??a? ?
  ep????? e??? f????? µ?a? t?ap???a? .
• ?a ?e????ta ?ep????? ?taµa? ?a? ?ep?????
  ???????? f????? e??a? a?e?a?t?ta d??t?
• ?(?)4/521/13, ?(?) 26/521/2
• ?(?,?) ?(ep????? ???????? ?taµa?) 2/52 1/26
• ?p?te ?(?,?) ?(?)?(?)
• ?p?s?? ?(??)2/261/13?(?), ?a?
  ?(??)2/41/2?(?)

9
2? ?a?ade??µa stat?st???? a?e?a?t?s?a?

• St?? ???? S 1,2,3,4,5,6 t?? ap?te?esµat?? t??
  ????? ?a???? ??????µe ta ?e????ta ?i lt 3 ?a?
  ? i a?t???.
• ???a? ?(?)2/61/3 ?a? ?(?)3/6 1/2
• ? ?(?,?) ?(i2) 1/6 ?(?) ?(?)
• ?p?s?? ?(??) 1/3 ?(?) ?a? ?(??) 1/2 ?(?)
• S?µe??te?? ?t? t? ?e????? G i ? 3 de? e??a?
  a?e?a?t?t? t?? ? (??at???)

10
Te???µa ?????? ???a??t?ta?

• ?? ta ?e????ta ?i, i1,2,n ap?te???? ??a
  d?aµe??sµ? t?? ????? ap?te?esµat?? S, d??ad? a?
• ???, a? ??a t? ?e????? ? e???µe t?? ?p? s??????
  p??a??t?te? ?(??i), i1,2,,n t?te µp????µe ?a
  ß???µe t?? p??a??t?ta ?(?) µes? t?? ?e???µat??
  t?? ?????? p??a??t?ta?

11
?a?ade??µa efa?µ???? t?? Te???µat?? t?? ??????
???a??t?ta?

• Te??e?ste t?? ???? ap?te?esµat?? S p?? p????pte?
  ap? t? ????µ? e??? ?a????, ?a? ta ?e????ta ?i
  i.
• ?a ?i ap?te???? ??a d?aµe??sµ? t?? ????? S
• Te??e?ste t? ?e????? ?a?t?? ap?te?esµa ?a?
  est? Q t? ap?te?esµa e??? pe??aµat??. ? ?(?)
  ß??s?eta? ?? e???

12
?a???a? t?? Bayes

• ? ?a???a? t?? Bayes d??e? t?? ?p? s??????
  p??a??t?ta ?(?i?) a? ?e???µe t?? ?p? s??????
  p??a??t?te? ?(??i) µes? t?? s?es??
• ?a?ade??µa G?a t? p??????µe?? pa?ade??µa
  ß??s???µe t?? ?(?2?) ?? e???

13
?s??s?

• Se µ?a p??? t?e?? µa??e? a?t?????t??, A, B and C
  ?ate???? t? 20, 30 ?a? 50 t?? a???a?,
  a?t?st???a.
• ? p??a??t?ta ?a ??e?as?e? e?a aµa?? ep?s?e?? t??
  p??t? ????? ?????f???a? t?? e??a? 5, 10 ?a?
  15, a?t?st???a.
• (a) ???a e??a? ? p??a??t?ta ep?s?e??? e???
  aµa???? t?? p??t? ????? ?????f???a? t????
• (b) ?? e?a aµa?? e?e? a?a??? ep?s?e??? t?? p??t?
  ????? p??a e??a? ? p??a??t?ta ?a e??a? µa??a? ??

14
?pa?t?s?
15
?fa?µ??? st?? ep????????e?

• ?etad?d??ta? s?µata ?i µe p??a??t?te? P(Ei).
• St?? ?e?t? ?aµßa?eta? t? s?µa R.
• ?p? µet??se?? e???µe ß?e? t?? p??a??t?te?
  ?(REi).
• ?a? e?d?afe???? ?? p??a??t?te? P(EiR).
• ??? ß??s???µe t?? P(EiR)??
• ?a???a? Bayes

16
???a?e? ?etaß??te? (rv - random variables)

• ??a t??a?a µetaß??t? X(s) e??a? µ?a p?a?µat???
  s??a?t?s? µe ped?? ???sµ?? t?? ???? t?? ?e????t??
  S, s ? S.
• ??a t??a?a µetaß??t? µp??e? ?a e??a?
• ??a???t?, ?
• S??e???
• ??a t??a?a µetaß??t? µp??e? ?a pe????afe?
• ?e t? s?µß??? t??, p.?. t? ? (pa?t?te ?efa?a??)
• ?e t?? pe????? t?µ?? t?? p.?. ? ? ?
• ?e t?? pe????af? t?? ?ata??µ?? t?? t?µ?? t?? x
  (?? t?µe? p?? pa???e? ? µetaß??t? s?µß??????ta?
  µe µ???? ??aµµa)
• ? s?es? ?x s?µß????e? t? ?t? ? t??a?a µetaß??t?
  ? p??e t?? t?µ? x

17
S??a?t?s? ?ata??µ?? p??a??t?ta? (PDF)

• ???µa?eta? ?a? s??a?t?s? a????st???? ?ata??µ??
  (Cumulative Distribution Function CDF)
• ???sµ?? FX(x) F(x) P(X ? x) Ps?S
  X(s)?x
• ?d??t?te?
• ? F(x) e??a? µ???t??a µ? a??a??µe??
• d??ad? F(a) F(b) a? a b
• F(-?) 0
• F(?) 1
• P(a lt X ? b) F(b) F(a)
• ??????t? ? CDF pe????afe? p????? t?? ?ata??µ?
  t?µ?? µ?a? t??a?a? µetaß??t??, ???s?µ?p??e?ta?
  s????este?a ? pdf ? pmf

18
S??a?t?s? ?????t?ta? ???a??t?ta? (pdf)

• ???sµ?? fX(x) dFX(x) /dx ? f(x) dF(x) /dx
• ? pdf pa??sta?e? t?? ???µ? a???s?? t?? CDF ? t?
  p?s? p??a?? e??a? ?a ?aße? ? X t?? t?µ? x
• ?d??t?te?
• f(x) ? 0
• ?
• ? f(x)dx 1,
• -? b
• P(a lt X ? b) ? f(x)dx F(b) F(a)
• x a
• F(x) ? f(s)ds
• -8

19
??aµe??µe?e? t?µe? (Expected values)

• ?? a?aµe??µe?e? t?µe? e??a? e?a? s??t?µ?? t??p??
  (µe?????) pe????af?? µ?a? t??a?a? µetaß??t?? X
• ?? p?? sp??da?e? e??a? ?
• ? µes? t?µ? ?(?) mX ? xf(x)dx
• 
  - ? ?
• H µetaß??t?t?ta s?2 E(X mX 2) ? (x mX
  )2 f(x)dx
• 
  - ?
• H s? ???µa?eta? t?p??? ap????s?
• ? ?p?????sµ?? t?? a?aµe??µe??? t?µ?? ???eta? µe
  a?a???? t??p? ?a? ??a ?p??ad?p?te s??a?t?s? g(X)
  t?? ?
• ?
• ?g(X) ? g(x)f(x)dx
• - ?

20
?d??t?te? µes?? t?µ?? ?a? µetaß??t?t?ta?

• ? µes? t?µ? e??a? ??a s?????? ??a µet?? t?? µes??
  t?µ?? t?? t?µ?? p?? pa??e? ? r.v. se µe?a??
  a???µ? pe??aµat??
• ?cX cEX
• Ec c
• EXc EXc ?p?? c sta?e?a
• H µetaß??t?t?ta e??a? ??a µet?? t?? d?asp??a? t??
  t?µ?? t?? r.v. ???? ap? t?? µes? t?µ?
• s?2 VARX ?(? mX)2
• VAR(cX) c2 VAR(X)
• VAR(c) 0
• VAR(Xc) VAR (X)

21
???s?t?ta Chebyshev

• ?st? ? t??a?a µetaß??t? µe µes? t?µ? mX ?a?
  µetaß??t?t?ta s?2
• ??te ??a ?a?e d, P(X - mX ? d) ? s?2 / d2
• ?? µe?e??? t?? µetaß??t?t?ta? ?a?????e? t? t??p?
  p?? ?ata?eµ??ta? ?? t?µe? t?? ???? ap? t?? µes?
  t?µ? t??
• ?? ???? p?? ?a?????eta? ap? t?? a??s?t?ta
  Chebyshev ???s?µ?p??e?ta? ??a t?? p??sd????sµ?
  t?? d?ast?µat?? eµp?st?s???? st?? t?µe? µ?a?
  p??s?µ???s??.

22
1? ?a?ade??µa ?µ???µ??f? ?ata??µ?

• ? 0.1, 0 ? x ? 10
• f(x) ?
• ? 0, a????

?e t? µetaß??t? a?t? pa??sta???µe t?? a???st?
fas? e??? ?µ?t???e?d??? s?µat?? µeta?? 0 ?a?
2p ? p ?a? p
f(x)
0.1
x
0
10
??a s??e??? t??a?a µetaß??t? e?e? ?µ???µ??f?
?ata??µ? µeta?? a ?a? b, a? pa???e? t?µe? µe ?s?
p??a??t?ta se d?ast?µata µe ?s? µ????.
23
1o ?a?ade??µa (s??e?e?a)

• ?es? t?µ?

• ?etaß??t?t?ta

• ?p?????sµ?? p??a??t?ta?

24
2? ?a?ade??µa Gaussian pdf?a?????? ?ata??µ?
N(mX, s?2)

s
N(0,1)
? p?? sp??da?a ?a? ????? rv. ? ?e?µ????
????ß?? e?e? ?a?????? ?ata??µ?
??a Gaussian t??a?a µetaß??t? ?a?????eta? p?????
ap? t?? µes? t?µ? ?a? t?? µetaß??t?t?ta t?? (?
t?? t?p??? ap????s?).
25
??a t??ep????????a?? s?st?µa µe Gaussian ????ß?
S ? ?a
lt
RSN
R 0??
gt
??µp??
?e?t??

N ?(0,s2)

• ? p??a??t?ta ?a ?a?e? sfa?µa ? de?t?? ?ta?
  ste??eta? t? S-a (?p?te t? ?aµßa??µe?? s?µa
  e??a? t? R-aN t? ?p??? e?e? ?ata??µ? ?(-a,sn) )
  e??a?

26
? s??a?t?s? sfa?µat?? Q-function

• ? s??a?t?s? sfa?µat?? e??a? ? t?p???? t??p??
  e?f?as?? t?? p??a??t?ta? sfa?µat?? se ??e?st?
  µ??f?

N(0,1)

• A???µ?t???? ?p?????sµ?? t?? s??a?t?s?? Q

??a x ? 3
27
? s??a?t?s? Q ?a? ? p??s????s? t??
x
28
3? ?a?ade??µa- Rayleigh pdf

• ?st?

?p?? ?? ? ?a? ? e??a? Gaussian r.v. µe µes?
t?µ? 0 ?a? µetaß??t?t?ta s2

• H R e??a? µ?a t??a?a µetaß??t? µe ?ata??µ?
  Rayleigh

• H Rayleigh pdf ???s?µ?p??e?ta? s???a ??a
  t??
• p??s?µ???s? t?? fa???µe??? t?? d?a?e??e??
  (fading)
• ?ta? de? e???µe s?µa ?pt???? epaf?? se µ?a
  as??µat?
• s??des? a??a s?µata ap? p???ap?e?
  d??de?se??

29
? Rayleigh pdf
30
S??a?t?se?? µa?a? p??a??t?ta?Probability Mass
Functions (pmf)

• ??a d?a???t? t??a?a µetaß??t? µp??e? ?a
  pe????afe? µe pdf a? ep?t?e???µe t?? ???s?
  ????st???? s??a?t?se??
• S?????? ?µ?? ???s?µ?p????µe t?? s??a?t?se?? µa?a?
  p??a??t?ta? (pmf)
• p(x) P(X x)
• ??e? ?d??t?te? a?t?st???e? t?? pdf, d??.
• p(x) ? 0
• S p(x) 1
• P(a ? X ? b)

31
?ese? t?µe? d?a???t?? t??a??? µetaß??t??

• G?a t?? d?a???te? t??a?e? µetaß??te? e???µe

32
?a?ade??µa 1 ??ad??? ?ata??µ?

• ???s?µ?p??e?ta? s????tata ??a t?? pa?astas?
  d?ad???? ded?µe???
• ?es? t?µ?
• ?etaß??t?t?ta
• ?? ?? ? ?a? ? e??a? a?e?a?t?te? d?ad??e? t??a?e?
  µetaß??te?,
• t?te pXY(0,0) pX(0) pY(0) ½ ½ 1/4

33
?a?ade??µa 2 ?????µ??? ?ata??µ?

• ??

?p?? ?? ?i, i1,2,,n
e??a? a?e?a?t?te?
d?ad??e? t??a?e? µetaß??te? µe

• t?te ? ?ata??µ? t?? Y e??a?

• ?es? t?µ? mY n p
• ?etaß??t?t?ta

34
?a?ade??µa 2 ?????µ??? ?ata??µ? (2)

• ?p??este ?t? e?peµp??µe µ?a a???????a ap? 31 bits
  ??d???p???µe?? µe ??d??a d?????s?? e?? ?a? 3
  ?a???
• ?? ? p??a??t?ta sfa?µat?? e??? bit e??a? p0.001
  p??a e??a? ? p??a??t?ta ?a ??f?e? ? a???????a µe
  sfa?µa??
• P(esfa?µe?? a???????a) 1- P(???? ????
  a???????a?)

• ?? de? ???s?µ?p????e? ? ??d??a? d?????s??
  ?a??? ?
• p??a??t?ta sfa?µat?? e??a?
• 1 (1-0.001)31 0.0305 3 ? 10-2

35
????ap?e? t??a?e? µetaß??te?

• ?st?sa? ?? r.v. ? ?a? Y p?? ??????ta? st?? ?d??
  ???? de??µat?? S. H ap? ?????? a????st???
  s??a?t?s? ?ata??µ?? (joint cdf) ????eta? ??
• FX,Y(x,y) P(X x, Y
  y)
• ?s? S X(s) x, Y(s) y
• e?? ? ap? ?????? s??a?t?s? ?ata??µ?? p??a??t?ta?
  (joint pdf) ????eta? ??

36
????ap?e? t??a?e? µetaß??te? (s??e?.)

• ?? ???a?e? (marginal) CDFs ?a? pdfs t?? ? ?a? Y
  e??a? ??
• ?p????µe ep?s?? ?a ???s??µe t?? ?p? s??????
  (conditional) pdf fXY(xy) fX,Y(x,y)/f?(y)
  e?? ??a stat?st??a a?e?a?t?te? r.v. e???µe
  fX,Y(x,y) fX(x) fY(y)
• ?µ???? µp????µe ?a ???s??µe s??a?t?se?? t?? d??
  µetaß??t?? g(?,?) ?a? ?a ?p?????s??µe t?? µese?
  t?µe? t???, ?p?? e??a? ? s?µµetaß??t?t?ta
  (covariance)
• COV(X,Y) s?,?2 ?(?-mX)(Y-mY)
  ???-mXmY
• ?? ?? ? ?a? ? e??a? a?e?a?t?te? t?te s?,?2 0
• To a?t?st??f? de? ?s??e? pa?a µ??? ??a Gaussian
  r.v.

37
?p? ?????? Gaussian µetaß??te?

• ? ap? ?????? pdf d?? ap? ?????? Gaussian
  µetaß??t?? e??a? ?
• ?p?? s?,?2 ?(?-mX)(Y-mY)
• a? s?,? 0 t?te fX,Y(x,y) fX(x)fY(y) gt X,Y
  a?e?a?t?te?
• d??. ??a Gaussian r.v a?e?a?t?s?a ? µ?de????
  s?s?et?s?
• ? t?p?? µp??e? ?a epe?ta?e? se n ap? ??????
  Gaussian µetaß??te?

38
?????sµata t??a??? µetaß??t??

• ?? e???µe µ?a a???????a n t??a??? µetaß??t??
  (?1, ?2,,?n) µe ßas??a t?? ?d?e? ?d??t?te?, t?
  µes? a????sµa t??? a?aµe?eta? ?a e?e? ????te??
  t??a?a s?µpe??f??a ap? t?? ???e µetaß??t?.
• ? ??µ?? t?? ?e?a??? ????µ?? ?a? t? ?e?t????
  ???a?? Te???µa ap?te???? t??? µa??µat???
  d?at?p?s? a?t?? t?? ?e????t??.

39
?s?e??? ??µ?? t?? ?e?a??? ????µ??

• ?? ?? t??a?e? µetaß??te? ?1, ?2,,?n e??a?
  as?s?et?ste? µe µese? t?µe? ?se? µe mX ?a?
  µetaß??t?t?te? ?se? µe s?2 lt8 t?te ??a ???e e gt
  0 e???µe
• ???ad? ? µes?? ???? t?? a????sµat?? t??
  µetaß??t?? s??????e? (?? p??? t?? p??a??t?ta)
  st?? ????? µes? t?µ?

40
?e?t???? ???a?? ?e???µa

• ?? ?e?t???? ???a?? Te???µa (Central Limit Theorem
  CLT) pe????afe? t?? ?ata??µ? t?? µes?? t?µ??
  t?? a????sµat?? µe?a??? p?????? t??a???
  µetaß??t??.
• ?? a?e?a?t?te? t??a?e? µetaß??te? ?1, ?2,..., ??
  e???? t?? ?d?a pdf µe µes? t?µ? 0 ?a?
  µetaß??t?t?ta s
• ??????µe t?? r.v.
• ?a??? t? ? ? ? ? ?ata??µ? t?? ? te??e? p??? t??
  ?a?????? (Gaussian) ?ata??µ? ?(0,s2/?))
• St?? p?a??, t? fa???µe?? ???eta? eµfa?e? a??µa
  ?a? ??a ?10
• ? ?e?µ???? ????ß?? p???a?e?ta? ap? t?? t??a?a
  ????s? t?? (s?ed?? ape???? t? p?????)
  ??e?t??????. ?ata s??epe?a µp??e? ?a ?e????e? µe
  µe?a?? a???ße?a ?t? ? ?ata??µ? t?? ?e?µ????
  ????ß?? e??a? Gaussian.

41
?a?ade??µa ??a t? ?e?t???? ???a?? ?e???µa
µ0, s1
?2
?5
N10
?10
42
?a?ade??µa

• http//www.jhu.edu/virtlab/stats/Stats.html