Introduction to Probability - Chapter 8 docx

... large to take n to get a desired accuracy. ✷ 8. 1. DISCRETE RANDOM VARIABLES 309 0 0.2 0.4 0.6 0 .8 1 0 0.02 0.04 0.06 0. 08 0.1 0 0.2 0.4 0.6 0 .8 1 0 0.02 0.04 0.06 0. 08 0 0.2 0.4 0.6 0 .8 1 0 0.02 0.04 0.06 0. 08 0.1 0.12 0.14 0 ... 1.00000 200 .15730 .50000 300 . 083 26 .33333 400 .04550 .25000 500 .02535 .20000 600 .01431 .16667 700 .0 081 5 .14 286 80 0 .004 68 .12500 900...

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... 185 0 .8 5000 2532 3.1596 Smith, 185 5 .6 3204 12 18. 5 3.1553 De Morgan, c. 186 0 1.0 600 382 .5 3.137 Fox, 186 4 .75 1030 489 3.1595 Lazzerini, 1901 .83 34 08 180 8 3.1415929 Reina, 1925 .5419 2520 86 9 ... Gauthier-Villars, 188 9). 4 E. T. Jaynes, “The Well-Posed Problem,” in Papers on Probability, Statistics and Statistical Physics, R. D. Rosencrantz, ed. (Dordrecht: D. Reidel, 1 983...

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... 5 3357 6 289 2 7 3657 8 3025 9 3362 10 2 985 11 31 38 12 3043 13 2690 14 2423 15 2556 16 2456 17 2479 18 2276 19 2304 20 1971 21 2543 22 26 78 23 2729 24 2414 25 2616 26 2426 27 2 381 28 2059 29 2039 ... 2729 24 2414 25 2616 26 2426 27 2 381 28 2059 29 2039 30 22 98 31 2 081 32 15 08 33 188 7 34 1463 35 1594 36 1354 37 1049 38 1165 39 12 48 40 1493 41 1322 42 1423 43 1207 44 12...

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... PROBABILITIES 3 .203309 .762057 .151121 .62 386 8 .932052 .4151 78 .716719 .967412 .069664 .670 982 .352320 .049723 .750216 . 784 810 . 089 734 .966730 .9467 08 . 380 365 .027 381 .900794 Table 1.1: Sample output ... 35 40 -1 0 -8 -6 -4 -2 2 4 6 8 10 Figure 1.1: Peter’s winnings in 40 plays of heads or tails. One can understand this calculation as follows: The probability...

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... PERMUTATIONS 79 Number of people Probability that all birthdays are different 10 .88 305 18 20 . 588 5616 30 .293 683 8 40 .1 087 682 50 .0296264 60 .00 587 73 70 .00 084 04 80 .000 085 7 90 .0000062 100 .0000003 Table ... =  1234 2143  , indicating that a 1 went to a 2 , a 2 to a 1 , a 3 to a 4 , and a 4 to a 3 . Ifwealwayschoosethetoprowtobe1234then, to prescribe the permuta...

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... Gauthier-Uillars, 188 8. 24 N. T. Gridgeman, Letter, American Statistician, 21 (1967), pgs. 3 8- 3 9. 4.2. CONTINUOUS CONDITIONAL PROBABILITY 169 one is, in effect, tossing a biased coin with probability ... drawn is red? 4.3. PARADOXES 175 13 Write a program to allow you to compare the strategies play-the-winner and play-the-best-machine for the two-armed bandit problem of Exampl...

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... Frequency Relative Frequency Frequency 1 17 .17 1 681 .1 681 -2 17 .17 16 78 .16 78 3 16 .16 1626 .1626 -4 18 . 18 1696 .1696 5 16 .16 1 686 .1 686 -6 16 .16 1633 .1633 Table 6.1: Frequencies for dice ... constant probability of about 5 /8 of surviving for another decade. 8 For example, of the 64 people who survive to age 6, 5 /8 of 64 or 40 survive to 16, 5 /8 of thes...

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... total service time for 10 customers. Estimate the probability that X>22 minutes. 288 CHAPTER 7. SUMS OF RANDOM VARIABLES 20 40 60 80 100 120 140 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0. 08 20 ... series lengths: four-game, five-game, six-game, seven-game. (b) Using one of the distribution found in part (a), find the probability that his batting average exceeds .400 in a four-game s...

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... .4332 2.5 .49 38 3.5 .49 98 .6 .2257 1.6 .4452 2.6 .4953 3.6 .49 98 .7 .2 580 1.7 .4554 2.7 .4965 3.7 .4999 .8 . 288 1 1 .8 .4641 2 .8 .4974 3 .8 .4999 .9 .3159 1.9 .4713 2.9 .4 981 3.9 .5000 Figure 9.4: ... .4 987 .1 .03 98 1.1 .3643 2.1 . 482 1 3.1 .4990 .2 .0793 1.2 . 384 9 2.2 . 486 1 3.2 .4993 .3 .1179 1.3 .4032 2.3 . 489 3 3.3 .4995 .4 .1554 1.4 .4192 2.4 .49 18 3.4 .4997...

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... BRANCHING PROCESSES 383 Generation Probability of dying out 1.2 2 .312 3 . 385 203 4 .437116 5 .47 587 9 6 .50 587 8 7 .529713 8 .549035 9 .564949 10 .5 782 25 11 . 589 416 12 .5 989 31 Table 10.1: Probability of ... PROCESSES 387 Geometric p j Data Model 0 .2092 . 181 6 1 .2 584 .3666 2 .2360 .20 28 3 .1593 .1122 4 . 082 8 .0621 5 .0357 .0344 6 .0133 .0190 7 .0042 .0105 8 .0011 .0...

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