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???. 4-2. ??? ???????. . ????. ??????:?????????????????????????????????:???????????????????????????????. ????. ???? (random variable) : ????? O ?????? R ???????????????????????????????? (univariate random variable) ????????,??????????????,????????????????. ??????. ??????(discrete random variable) : ???????????????? (finite) ????????? (countably infinite) ? ??: ?????????????????,????????????: ???????????????????????????,??????????? .

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    1. ??? 4- 4-1

    2. ??? 4-2

    3. ???? ??????: ??????????? ????????? ??????? ??????: ??????????? ????????? ???? ???????

    4. ???? ???? (random variable) : ????? O ?????? R ???? ?????????????????? ?????????? (univariate random variable) ? ???????,??????????????,????????????????

    5. ?????? ??????(discrete random variable) : ???????????????? (finite) ????????? (countably infinite) ? ??: ?????????????????,?????????? ??: ???????????????????????????,???????????

    6. ???? X ???????????,??? b1,b2,…? ?? X ????? fx ? fX(bi ) = P({?:X(? ) = bi }), i = 1,2,3… ?????? fx ? 0 ? ???????? X ???????????????,???? X ?????? fx ?????????:

    7. ?????? ?????? (cumulative distribution function) :???? X ?????? R ??? [0 1] ???,????? FX(a) = P({?:X(? ) ? a}) ?????????????????????????????????(discrete distribution function) ? ?????????????????? ??????????????? ?????? FX ????? (step function) ,?? X ??????? b1,b2…..??????? (jump) ???????????????

    8. ????????? -- ??? ??????? (moment) ???????????? (?????????)?????????,??????????? ? fX ???????? X ?????,E ???????(expectation operator) ? E(X) ?? X ??? (mean) ???? (expected value) ,???????: ???????????????????? (weighted average),???????????? ????????,?????????

    9. ????????? -- ??? ????????:?????? a ? b, E (aX+b) = a E(X) + b ?????????????????,????????????? ???????????????????? ?????????????????????

    10. ????????? -- ??? ?????? X2 ????? X ???????? bi ????????: E(X2) ??? X ?????? (second moment): ????? ???????? (variability) ?????? ???????: ??????????????????????,???? X – E(X) ?????????????????: ????????????? (second central moment) ???? (variance), ??? var(X) ?

    11. ????????? -- ??? ???????: ?? X ???????????????,???????? ??????????????,?????????????????? ?????????????????????????: var(X) =E[X – E(X)]2 = E(X2) ? [E(X)]2 ???????????? (standard deviation) ? ? X ???????????? a ? b , var(aX+b) = var(aX) = a2 var(X) ???????????,???????????

    12. ????????? -- ??? ???????? (standardization): ???????????????????????? ??????? X ,? ?????????????????? ????? 0, ????? 1 ?

    13. ?????? -- ?? ???????? X1 ? X2 ????????: P({X1 = -2}) = P({X1 = 2}) = 0.5, P({X2 = -2}) = P({X2 = 4}) = 0.5. ??, E(X1) = (-2)(0.5) + (2)(0.5) = 0, E(X2) = (-2)(0.5) + (4)(0.5) = 1. E(X12) = (-2)2(0.5) + (4)2(0.5) = 10, E(X22) = (-2)2(0.5) + (2)2(0.5) = 4 . var(X1 ) = E(X12) – E(X1) 2 = 4 , var(X2 ) = E(X22) – E(X2) 2 = 9 .

    14. ?????? -- ?? ? X3=X1 + 2,? E(X3) = (0)(0.5) + (4)(0.5) = 2, E(X32)=8 var(X3 )= E(X32) – [E(X3)]2 = 4 .

    15. ??????? ????? (Bernoulli experiment): ??????????? ??:??????????? ??????? (Bernoulli random variable) X ?: X = 1, ????????, X = 0, ????????.

    16. ??????? ? p ?? P({ X=1 }), ??????????????: fX(b ; p) = P({ X = b }) = pb ? (1 ? p)(1?b),b = 0,1 fX ??????? (parameter) p,? p ???????? {X=1} ?????????? E(X) = 1 ? p + 0 ? (1 ? p) = p. var(X) = (1 ? p)2 ? p + p2 ? (1 ? p) = p(1 – p) ????????????:

    17. ?????? ?????? (continuous random variable ) ?????????? (continuous distribution function) ?????: ???? X ??????? FX ????? (continuous function),???????????????????(differentiable) ? ? X ????????,??????? a ,P({X=a})=0 ?? X ??????? a ?????? 0 ? ? X ????????,FX???? FX (a) = ?a-? fX(b) db ? ?? fX ? FX ????,?????????? fX ?????????? FX (a) ??????????????????????????????

    18. ?????? ? X ???????, fX (a) ? P({X=a}) ?????;?????? 0,??????? 0? ???????????????????,????????

    19. ????????? ????????????????,??????,????????? ?????????: ?????????? ?????????: ????????????? ? fx ?????????X ???(???)? E(X)? X ???????????,????????????????? X ????????????:

    20. ????????? ?????????????????????????????,???????????????,???????? ??????????????????????? ? E(X) ????,???? E(X) ???,?? X ???????? ???????????????(????) ,???????????

    21. ??????—?? ??? [r,s] ???????????????,????????: ??????: ??: ???:

    22. ???? ? q ??? 0 ? 1 ???????????? Fx ? q ???(quantile) ???? ??? d ?;? dq ?? q ???,? ?? q ???????????????,???????????????? ? d* ???????????,? d* ??????: ?????????????????????????????????????,????????

    23. ??????? ????? X ????????,??????? c > 0, P({|X ? E(X)| ? c}) ? var(X)/c2. ??:? k = 2,???? 3/4 = 75% ?????????????????? ??:? k = 3,???? 8/9 = 88.9% ??????????????????

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