随机过程-泊松过程

∙ Ïdk
0( 1, 2
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+ ℎ) − ℎ
0( 1, 2)
=−
0( 1, 2) +
∘(ℎ) . ℎ
-ℎ → 0
0( 1, 2) 2
=−
0 ( 1 , 2 ). 0( 1, 2) ( 2− 1) ,
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H®e>ŒÆ2009?a¬ïÄ) / Hený¢
«ã4Ò
¿
2009.10.13-2009.12: 4-7•( : 1-4!); 10+15-17+19-20•(o:1-4!) 2009.11.21-2010.01: 22-23•(8: 5-8!)
‘ÅL§ / Stochastic Processes /\ì
http://math.carleton.ca/∼tangjs
URL Reµhttp://here.is/tangjs E-mail: jiashant@yahoo.ca
H®e>ŒÆ nÆ
ÚOX
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This document is created by Jiashan TANG with NUPT on October 26, 2009: jiashant@yahoo.ca; http://math.carleton.ca/∼tangjs
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3
Poisson Processes
■ Poisson processes has a strong background. Let’s begin with “counting processes”. In our real world, we often encounter the following problems: (1) 3[0, ] žmS,>{oÅ • gê. (2) ,ÑÖXÚ3[0, ] žmS‡¦ÑÖ •<g. (3) Åì[0, ] žmSu) æ gê. (4) 3 ê i Ï & ¥, ® ? è & Ò § 3 D Ñž 3[0, ] ž m S u )Øè ‡ê. . This kind of problems has a common characteristic: • Ä[0, ] ž mS,a¯œu) gê§PŠ: ( )
This document is created by Jiashan TANG with NUPT on October 26, 2009: jiashant@yahoo.ca; http://math.carleton.ca/∼tangjs All right reserved.
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dЩ^‡ 0( 1, 1) = 1, )þ㕧 ( ( 2 − 1)) −1 − −1 ( 1 , 2 ) = ( − 1)!
=
− ( 2− 1) .
¦ ( 1 , 2 ), † þ ¡ Ó
This document is created by Jiashan TANG with NUPT on October 26, 2009: jiashant@yahoo.ca; http://math.carleton.ca/∼tangjs
ë•Öµ(1)•²À. ‘ÅL§n؆A^. ˜uŒÆч §2005c8 Sheldon M. Ross. Stochastic Processes. John Wiley & Sons, Inc., 1983
This document is created by Jiashan TANG with NUPT on October 26, 2009: jiashant@yahoo.ca; http://math.carleton.ca/∼tangjs All right reserved.
∙பைடு நூலகம்
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Proposition 19 (p38/31) Let { ( ), ≥ 0} be a Poisson process whose intensity is . Then for ∀ 2 ≥ 1 ≥ 0, increment ( 2) − ( 1) is a Poisson random variable with the parameter given by ( 2 − 1), i.e.: ( ( 2 )− ( 1 ) = ) = ( ( 2 − 1)) − ( 2− 1) , ! Proof. |^êÆ8B{\±y²§ ( 1, 2) = ( ( 2) − ( 1 ) = ),
( ( 2 + ℎ) − ( 2) = 1) ( ( 2) − ( 1) = − 1)+ ∑ ( 2) = ) ( ( 2) − ( 1) = =2 ( ( 2 + ℎ) − = = l ( 1, 2)
2
( 1, 2)(1 − ℎ + ∘(ℎ)) + −1( 1, 2)( ℎ + ∘(ℎ)) ∑ + − ( 1 , 2 ) ∘ (ℎ) =2 ( 1, 2) − ( 1 , 2 )ℎ + =−
This document is created by Jiashan TANG with NUPT on October 26, 2009: jiashant@yahoo.ca; http://math.carleton.ca/∼tangjs
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This document is created by Jiashan TANG with NUPT on October 26, 2009: jiashant@yahoo.ca; http://math.carleton.ca/∼tangjs
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Condition (3) L²3v « m S, Ä þ • k ˜ ‡ ‘ Å : u ), • = 3 v «mSu)ü‡±þ‘Å: Vdžk˜‡ ‘Å:u) VÇ' ´p á . ùÒ´`·‚•Ä ´˜« { ü œ /, ¢ S þ 3 ? ˜ ž • u ) ü ‡ ± þ ¯ ‡ 3 ¢ S ) ¹ ¥ ´ „ , Ä– êÆ .¥@•´4 Vǯ‡" ∙ † ÷ v ± þ^ ‡ Ñ tL § ƒ éA ‘ Å :6 • ¡ • {ü Ñt6. 'uÑtL§ ©Ù§dc¡• ÕáOþL§ k•‘é Ü©Ù d § ˜ ‘ © Ù 9 O þ © Ù ( ½, é Ñ t L § ù ˜ A ÏL§5`, Ï• ( (0) = 0) = 1 9Oþ þ!5, •‡• § ˜‘©ÙÒŒ (½Ùk•‘©Ù¼ê§Ïde¡ ·K(é£ ãÑtL§ ©Ù)´¿© .
=2 2
( 1) = 0) ( ( 2 + ℎ) − −
1( 2, 2
− ℎ + ∘(ℎ)).
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This document is created by Jiashan TANG with NUPT on October 26, 2009: jiashant@yahoo.ca; http://math.carleton.ca/∼tangjs
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3.1
Definition
Definition 18 (p37/31, Def 3.3) A counting process { ( ), ≥ 0} is called a Poisson process if it satisfies the following conditions: (1) ( (0) = 0) = 1. (2) Increments is homogenous and independent. (3) ( ( + ℎ) − ( ) = 1) = ℎ + ∘(ℎ), ( ( + ℎ) − ( ) ≥ 2) = ∘(ℎ), where > 0 is called the intensity of ( ). þã½Â¥ˆ^‡ ¿Â: ^‡(1) •´•ïÄ •B¤‰ ˜«5½. ^‡(2) ´Äu,«†* b , ˜•¡, ˆ‘Å: u)´ƒ pÕá , u´3؃-U «mS‘Å: u)‡ê´ƒpÕá §= ( ) kÕáOþ. ,˜•¡, ‘Å: u)´²- , =3˜ ½ •Ý«mS§‘Å: u)‡ê ÚO5Æ5•†«m•Ý k' †å©žmÃ', Ïd ( ) „´þ!ÕáOþL§"
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á 8 ¹
Ch 1 VÇØÄ: Ch 2 ‘ÅL§Ä Ch 3 ÑtL§ Ch 4 ê ‰Åó ‰Åó Vg
Ch 5 ëYëêê Ch 6 ²-‘ÅL§
Ch 7 ²-L§ Ì©Û áµ4gu. ‘ÅL§9ÙA^(1n‡). p (2)Û(É È. ‘ÅL§(1˜‡). ¥IÚOч ˜Ñ‡ , 2004.7 2005c8 (15g<M)
This document is created by Jiashan TANG with NUPT on October 26, 2009: jiashant@yahoo.ca; http://math.carleton.ca/∼tangjs All right reserved.
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∙ ½ž, w, ( ) ´˜‡‘ÅCþ ∙ e ≥ 0 Ø ½ , = ? 1 Ø m ä * , u ´ { ( ) , ≥ 0} ´ ˜‡‘ÅL§" ∙ { ( ), ≥ 0} ´þãy– ˜‡êÆ ."ù´˜‡ëêë YG lÑ ‘ÅL§"=§ G ˜m = {0, 1, 2 ⋅ ⋅ ⋅ }, § ¼êÑ´4O F¼ê. ■ù ‘ÅL§{ ( ), ≥ 0} ¡•PêL§. ¦ ( ) Š u ) C ž • , = 1 , 2, ⋅ ⋅ ⋅ • Ò ´ , a ¯ œ u ) ž•, w,, ù ž••´‘Å , ¡Š‘Å:" ( ) Ò´3[0, ] žmm…S‘Å: ‡ê. Remark: e3,ž• Óžkü‡Óa¯œu), K n)• kü‡‘Å:-Ü(ü‡‘Å:±þ œ¹aq•Ä). The Poisson process introduced bellow is one class of important counting processes.
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∙ n ( 1, 2 + ℎ) = ( ( 2 + ℎ) − = = ( ( 2 + ℎ) − ( ( 2 + ℎ) −
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( 1) = ) ( 2) + ( 2) − ( 1) = ) ( 1) = )+ − ) ( 2) = 0) ( ( 2) −
2
≥ 1, = 0, 1, 2, ⋅ ⋅ ⋅ .
= 0, 1, 2, ⋅ ⋅ ⋅ .
Äk¦) 0( 1, 2), d½Â•
0( 1, 2
+ ℎ) = = = = =
( ( 2 + ℎ) − ( ( 2) −
0 ( 1 , 2 )(1 0 ( 1 , 2 )(1 0( 1, 2) 0( 2, 2
( 1) = 0) ( 2) = 0) ( 2, + ℎ)) + ℎ) + ℎ) − ∑∞