Chapter Three

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Chapter Three

• Introduction to Probability

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

• ??(Probability) ????????
• ??(Event)????????????????
• P(A)??????, ??A?????
• ???? ? ?? ? ??
• ?????????
• ?????A, 0 ?P(A) ?1
• P(Ã) 1 - P(A), ??P(Ã)?A??????
• ???A, B??, ? P(A?B) P(A) P(B), ?P(A?B) 0

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

• ?P(A?B) P(A) P(B), ???A?B??
• P(A?B) P(A) P(B) - P(A ? B)
• ????
• ????
• (Bayesian)
• Examples

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Ex(1)????A?B?????A????300???, ?????0.3,
B????2000???, ????0.4, ?????????,????????????,????
?A????????? (2)?????A?B???????????C,
????l000???, ????0.1, ??A,B????, ???????????,
???A?????????
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EX?????????????????????,??????????,????????? ??
?????0.7 ???????1.4 ??????13 ?????????0.8
??????????1.3 ??????????1.5 ???,???????llp
(1)???????????????????? (2)????????????,?????????
???????? (3)??????????????????????,???????
????????? (4)??????????????
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Mean and Variance of a Continuous Random Variable
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??????????
??????????

• ?????(m ) ???????? E(X)
• ?????(s 2) ???????? V(X)
• Examples

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

• ?????
• ????
• ?????
• ?????
• ?????
• ????
• ????
• ????

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????(Binomial Distribution)
??????????

• ????
• n?????????, ????????
• ???????????(S)???(F)??
• ?????? P(S) p
• ?????? P(F) q
• ???? (x) n?????????
• ????
• m E(X) np
• V(X) npq
• Examples

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EX ??????????500?????,?????,????????l0???,
????????????, ????, ???????? (1)????????l0????,??
????????????? (2)??????????20?,??????????????
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?????
??????????

• ???? ??????, ???????????, ??????
• ???? (x) n?????????
• ????
• Examples

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?????(Poisson)
??????????

• ????
• ???????, ???????????
• ???????????????? l
• ?????????????????????
• ?????????, ??????????????
• ???? (x) ?????????, ???????
• ????
• m E(X) l, V(X) l
• ??? XPoisson(l1), YPoisson(l2),
• ? XYPoisson(l1 l2)
• Examples

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????????????,???????,?????,?????,??????????. ????
??????,????????????????3????,????????,????????????
??????????????
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????(Normal)
??????????

• ????
• E(X) m, V(X) s 2
• P(m-sltXltms) 0.683
• P(m-2sltXltm2s) 0.954
• P(m-3sltXltm3s) 0.997
• ???
• ?????
• Examples

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Standardization
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Standardization - Cont.
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? ??????????,??????8.8750.005?,
????????XN(8.873,0.0042),?? (1)??????????????? (
2)??????10?,??????????????? (3)??????yN(8.875,0.0
042),????????????????? (4)??????yN(8.875,0.0022),
?????????????????
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??????????15?0.3??,????,?????,????????N(14.9,0.04)
,???????N(14.?,0.01),?? (1)?????????????????? (2)
????????????????????? (3)????????????????????
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????(Exponential)
??????????

• ???????, ??????????????, ???????????????
• ???? (x) ?????????, ??????????
• ????
• m E(X) 1/l, V(X) 1/l2
• Lack of memory
• Examples

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??????????,????????,????10.3??,?? (1)??????????
(2)???????????????????????
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????(Weibull)
??????????

• ???? (t) ??
• ????
• m E(X) l, V(X) 1/l2
• Examples

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EX You are conducting a reliability analysis for
a new product. Based on prior testing with a
similar product.you believe the weibull
failure-time distribution, with parameter?0.20
per year, applies. But you have no basis for
establishing b. Compute at a time span of 5
years (1) the failure rate (2)the survival
probability assuming that (a) b 0.5 (b) b
1.2, and (c) b 2.0.
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Normal Approximation to Binomial and Poisson
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Covariance and Correlation