Chapter 5 Special Discrete Distributions. Wen-Guey Tzeng Computer Science Department National Chiao University

Transcription

1 Chatr 5 Scal Dscrt Dstrbutos W-Guy Tzg Comutr Scc Dartmt Natoal Chao Uvrsty

2 Why study scal radom varabls Thy aar frqutly thory, alcatos, statstcs, scc, grg, fac, tc. For aml, Th umbr of customrs a rod of tm. Th g dstrbuto gtcs Lf tm of a dvc 2

3 Broull radom varabl s a Broull r.v. wth aramtr f S={0, } ad ts robablty mass fucto s othrws q 0 0 : succss, 0: falur 3

4 E = 0 P=0+ P== E 2 = 0 2 P=0 + 2 P== Var = E 2 E 2 = 2 = - 4

5 Toss a far d. Th vt of obtag 4, 6 s calld a succss, ad th vt of obtag, 2, 3, or 5 s calld a falur. Lt dot whthr a toss s a succss or ot 0 f 4 or 6 s othrws obtad Th, s a Broull radom varabl wth aramtr =/3. 5

6 Bomal radom varabl Broull trals, all wth robablty of succss, ar rformd ddtly Lt b th umbr of succsss ths trals s a bomal r.v., dotd as B, lswhr,...,,, f P 6

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9 A rstaurat srvs 8 trs of fsh, 2 of bf, ad 0 of oultry. If customrs slct from ths trs radomly, what s th robablty that 2 of th t four customrs ordr fsh trs? Sol: ~ B4, 8/30 ad calculat P=2 9

10 Ectato E ] [!!!!!!!!! 0

11 Lt b th rsult of th th Broull tral wth succss robablty = : th umbr of succsss of trals s B, E = E = Varato 2 Comut E 2 = Var = E 2 2 = - Sc, 2,, ar ddt Var = Var = Var + Var 2 + Var =-

12 Posso radom varabl A radom varabl that couts th umbr of occurrcs of a vt a gv ut tm, dstac, ara, or volum Th umbr of alha artcl mssos of a radoactv matral a hour Th umbr of brths a hostal a day Th umbr of arrval customrs a surmarkt btw 2m ad 3m Th rat s costat or statoary ay gv ut 2

13 W chck 2,... 0,, for! P!! 0 0 3

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16 A Posso r.v. wth s a aromato of B, s small s larg = s modrat 80 studts my class Each studt would mss a class wth robablty 0.05 ddtly Th umbr of abst studts a class s B80, 0.05 W cosdr th avrag rat of abst studts a class s =800.05=4 W ca trat th umbr of abst studts class as a Posso radom varabl wth =4 6

17 ! modrat, small for larg,! 2!!! P P 7

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19 j P E j j!!! Ectato & varac

20 ! 2!! ] [ j P E j j 20

21 E[ 2 ] E[ ] 2 Var E[ 2 ] E 2 2 E[ 2 ] 2

22 Lt b th umbr of wg tckts amog th Marylad lottry tckts sold Baltmor durg o wk. s a bomal radom varabl bcaus : th total umbr of tckts sold Baltmor s larg P: th robablty that a tckt ws s small : th avrag umbr of wg tckts s modrat s aromatly a Posso r.v. wth = P=0 = - 0 /0! 22

23 Evry wk th avrag umbr of wrog-umbr ho calls rcvd by a crta mal-ordr hous s 7. What s th robablty that thy wll rcv 2 wrog calls tomorrow at last o wrog call tomorrow? Sol: E = 7/7 =. So, ~ Posso P! a P 2 /2! 0.8 b P P

24 Gomtrc radom varabl Lt b th umbr of rmts utl th st succss occurs ad lt th robablty of succss, 0<<. P==- -, =, 2, 3, E = / Var = -/ 2 24

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26 From a ordary dck of 52 cards w draw cards at radom, wth rlacmt, ad succssvly utl a ac s draw. What s th robablty that at last 0 draws ar dd? Sol: ~ Go/3 P

27 Ngatv bomal radom varabl Lt b th umbr of rmts utl th r th succss occurs ad lt th robablty of succss, 0<<. P==C-, r- r - -r, =r, r+, E = r/ Var = r-/ 2 27

28 Sharo ad A lay a srs of backgammo gams utl o of thm ws fv gams. Suos that th gams ar ddt ad th robablty that Sharo ws a gam s Fd th robablty that th srs ds sv gams. If th srs ds sv gams, what s th robablty that Sharo ws? 28

29 Sol: Lt b th umbr of gams utl Sharo ws 5 gams. Lt Y b th umbr of gams utl A ws 5 gams. ~ NB5, 0.58 ad Y ~ NB5, 0.42 PEd 7 th gam = P=7+PY=7= =0.24 Lt A b th vt that Sharo ws ad B b th vt that th srs ds 7 gams. PA B=PAB/PB =P=7/[P=7+PY=7] =0.7/0.24=0.7 29