Introduction to Probability - Chapter 3 ppt

... {a 1 ,a 2 ,a 3 ,a 4 } can be written in the form σ =  1 234 21 43  , indicating that a 1 went to a 2 , a 2 to a 1 , a 3 to a 4 , and a 4 to a 3 . Ifwealwayschoosethetoprowtobe1 234 then, to prescribe ... ω 2 , ω 3 , and ω 5 . Each of these paths has the same probability p 2 q.Thusb (3, p,2)=3p 2 q. Considering all possible numbers of successes we have b (3, p,0) = q 3...

Ngày tải lên: 04/07/2014, 10:20

... vol. 204 (1 937 ), pp. 39 7 39 9. 7.1. SUMS OF DISCRETE RANDOM VARIABLES 287 = 1 36 · 1 6 = 1 216 , P (S 3 =4) = P(S 2 =3) P (X 3 =1)+P (S 2 =2)P (X 3 =2) = 2 36 · 1 6 + 1 36 · 1 6 = 3 216 , and ... (S 2 =8)=5 /36 , P (S 2 =9)=4 /36 , P (S 2 =10) =3/ 36, P (S 2 =11)=2 /36 , and P(S 2 =12)=1 /36 . The distribution for S 3 would then be the convolution of the distribution fo...

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... -5 0 00000000000 0-1 00 11111124979 730 0 233 420000000 0 100000000000 -5 0 210000000000 0 33 471117141111101625 130 0 00000000000 0-1 00 1221 131 00000 -5 0 00000000000 0-1 00 231 000000000 0 31 0000000000 50 100000000000 -5 0 34 4710119111214 131 0 ... BRANCHING PROCESSES 38 3 Generation Probability of dying out 1.2 2 .31 2 3 .38 52 03 4 . 437 116 5 .475879 6 .5...

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... (a) P =       34 512 30 2 /30 1 /30 41 /30 2 /30 0 502 /30 01 /3 100010 200001       . (b) N =   34 5 35 /32 4 /3 4 132 52 /32 7 /3   , t=   35 46 55   , B=   12 35 /94/9 41 /32 /3 52/97/9   . (c) ... (a)  4 1  13 10   52 10  =7. 23 × 10 −8 . (b)  4 1  3 2  13 4  13 3  13 3   52 10  = .044. (c) 4!  13 4  13 3  13 2 ...

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... DISCRETE PROBABILITIES 3 .2 033 09 .762057 .151121 .6 238 68 . 932 052 .415178 .716719 .967412 .069664 .670982 .35 232 0 .0497 23 .750216 .784810 .089 734 .966 730 .946708 .38 036 5 .02 738 1 .900794 Table 1.1: ... .5 13. Thus, it is more appropriate to assign a distribution function which assigns probability .5 13 to the outcome boy and probability .487 to the outcome girl tha...

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... 5000 2 532 3. 1596 Smith, 1855 .6 32 04 1218.5 3. 15 53 De Morgan, c.1860 1.0 600 38 2.5 3. 137 Fox, 1864 .75 1 030 489 3. 1595 Lazzerini, 1901 . 83 3408 1808 3. 1415929 Reina, 1925 .5419 2520 869 3. 1795 Table ... = 35 5/1 13: L = πP(E) 2 = 1 2  35 5 1 13  1808 34 08  = 5 6 = . 833 3 . Even with careful planning one would have to be extremely lucky to be able to sto...

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... precise below. ✷ 146 CHAPTER 4. CONDITIONAL PROBABILITY Number having The results Disease this disease ++ +– –+ –– d 1 32 15 2110 30 1 704 100 d 2 2125 39 6 132 1187 410 d 3 4660 510 35 68 73 509 Total 10000 Table ... Appendix C) to compute a con- ditional probability. The number 93, 7 53 in the table, corresponding to 4 0- year-old males, means that of all the males bor...

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... 2892 7 36 57 8 30 25 9 33 62 10 2985 11 31 38 12 30 43 13 2690 14 24 23 15 2556 16 2456 17 2479 18 2276 19 230 4 20 1971 21 25 43 22 2678 23 2729 24 2414 25 2616 26 2426 27 238 1 28 2059 29 2 039 30 2298 31 ... 2616 26 2426 27 238 1 28 2059 29 2 039 30 2298 31 2081 32 1508 33 1887 34 14 63 35 1594 36 135 4 37 1049 38 1165 39 1248 40 14 93 41 132 2 42 14 23 43 1207 44...

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... VARIANCE W L W L W L W L W L W L (2 ,3, 12) L 10 9 8 6 5 4 (7,11) W 1 /3 2 /3 2/5 3/ 5 5/11 6/11 5/11 6/11 2/5 3/ 5 1 /3 2 /3 2/9 1/12 1/9 5 /36 5 /36 1/9 1/12 1/9 1 /36 2 /36 2/45 3/ 45 25 /39 6 30 /39 6 25 /39 6 30 /39 6 2/45 3/ 45 1 /36 2 /36 Figure ... 36 possibilities. Thus, x = 31 36 · y. But when Huygens rolls he wins on 6 out of the 36 possible outcomes, and in th...

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... 8.2. CONTINUOUS RANDOM VARIABLES 31 9 n P (|S n /n|≥.1) Chebyshev 100 .31 731 1.00000 200 .15 730 .50000 30 0 .0 832 6 .33 333 400 .04550 .25000 500 .02 535 .20000 600 .01 431 .16667 700 .00815 .14286 800 .00468 ... x) to compute the exact probability that you estimated in Exercise 1. Compare the two results. 3 Write a program to toss a coin 10,000 times. Let S n be the number of...

Ngày tải lên: 04/07/2014, 10:20