4.??????????? Distribution Functions and Discrete Random Variables

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4.??????????? Distribution Functions and Discrete
Random Variables
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4.1 ????(Random variables)

• ?? ?????? X S?R ?????????? ??, ????? I R,
  ?????
• ???, ?? ?????
• ? ?????

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????(Random variables)

• ?????52???????3?(??????????).
• ? X ???????
• ? X ???????.

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????(Random variables)

• ??????? s,????? t,?x???????? ????????
• S(s,s,s), (t,s,s), (s,t,s), (s,s,t),
• (s,t,t), (t,s,t), (t,t,s),
  (t,t,t),
• by X(s,s,s)3, X(s,t,s)2, X(s,s,t)2,
  X(s,t,t)1, and so on.

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????(Random variables)

• P(X0)P((t,t,t))27/64
• P(X1)P((s,t,t),(t,s,t),(t,t,s))27/64
• P(X2)P((s,s,t),(s,t,s),(t,s,s))9/64
• P(X3)P((s,s,s))1/64
• ?????????
• P(Xi)C(13,i)C(39,3-i)/C(52,3)
• for i0,1,2,3.

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????(Random variables)

• ?????, ??????????1/90.
• ? X ???????????????? ? X ?????.

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????(Random variables)

• ? T?????? N?????. X ?????????????
• X ????? 1, 2, 3,

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4.2????(Distribution functions)

• ???? X ?????(random variable), ??? F ???
  ??? F(t)P(Xltt) ??? X ?????.

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????(Distribution functions)

• ?????????
• F ?????? ?, if tltu, then F(t)ltF(u).
• 2.
• 3.
• 4. F ???? ?, ?R???t, F(t)F(t)

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????(Distribution functions)

??X??? ?F??????? ??X??? ?F???????
X lt a F(a) a lt X lt b F(b) F(a)
X gt a 1 F(a) a lt X lt b F(b-) F(a)
X lt a F(a-) a lt X lt b F(b) F(a-)
X gt a 1 F(a-) a lt X lt b F(b-) F(a-)
X a F(a) F(a-)
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????(Distribution functions)

• ?????????X ?????

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????(Distribution functions)

• ??
• P(Xlt2)
• P(X2)
• P(1ltXlt3)
• P(Xgt3/2)
• P(X5/2)
• P(2ltXlt7)

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????(Distribution functions)

• ???????????1000?1030????????. ? X ????? ?? X
  ????????.
• Sol

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4.3 ??????(Discrete random variables)

• ?? ??????X????????,??????????(countable), X
  ????????.
• ???????
• finite set 0, 1, 2
• countable set 1, 2, 3, 4,
• uncountable set x x gt 0

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??????(Discrete random variables)

• ?? ????X ?????(probability function) p ???? x1,
  x2, x3, ???R to R???,??????.
• (a) p(x)0 if x x1, x2, x3,
• (b) p(xi)P(Xxi) and hence p(xi)gt0
• (c)

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??????(Discrete random variables)

• ??
• ??????????
• Sol

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4.4??????????( Expectations of discrete random
variables)

• ?? ????X??????
• ??p(x) ???????? A?????X ????
• ???????????E(X) ??.
• E(X) ??????(mean)
• ? X??????EX??, ? .

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??????????( Expectations of discrete random
variables)

• ?? ???? , ?????????????. ??????????K???, ????
  2k ?.
• ?? ?????, ?????????????????

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??????????( Expectations of discrete random
variables)

• ? ? X ???????. ? X ??????,???? 2, 4, 8, ?
  P(X2k)1/2k, k1, 2, 3,
• ??,
• ?????
• ?????????????????????.

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??????????( Expectations of discrete random
variables)

• ??4.1 ?? X ????????, ?, P(Xc)1, c???, ? EXc.
• ??4.2 ? g ??????,? g(X) ???????

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??????????( Expectations of discrete random
variables)

• ???? ? g1, g2, , gn????, ? ? a1, a2, , an ???.
  ?

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4.5?????????????(Variances and moments of
discrete random variables)

• ?? X????(Variance)
• X????(Standard deviation )

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?????????????

• ?? 4.3 Var(X) EX2 (EX)2
• ?? Var(X) E(X-EX)2
• EX2 2XEX (EX)2
• E(X2) 2EXEX (EX)2
• E(X2) (EX)2
• ?? (EX)2 lt EX2

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?????????????

• Ex 4.24 ??????????? (outcome )?,?????X???????
• Sol EX(123456)/67/2
• EX2(149162536)/691/6
• Var(X)91/6-(7/2)235/12

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?????????????

• ?? 4.4 Var(X)0 ????
• X ???????? ??? 1.
• ??4.5 Var(aXb)a2Var(X)

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?????????????

• Ex 4.25 EX2 and EX(X-4)5.
• Var(4X12)?
• ? EX2-4XEX2 4EX5
• so EX254x213
• Hence Var(X)EX2 (EX)2
• 
  13-229
• ??? 4.5
• Var(4X12)16x9144

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?????????????

• ?? ? w ?????. X ?Y???? w.
• ????? t gt 0 ?
• P(Y-wltt) lt P(X-wltt)
• ?? 4.6 ?? EXEYa. ? X ?Y???? a , ?
  Var(X)ltVar(Y)

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?????????????

• ?? ? c ????, ngt0 ???, ? rgt0 ????.

Eg(X) ??
E(Xn) E(Xr) E(X-c) E(X-c)n E(X-EX)n X? n ?? X? n ???? X? 1 ?? about c X? n ?? about c X? n ????
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4.6 ???????

• ?? ? X ??????????µ (mean )????(standard
  deviation ) s . ?X????????.
• ???