Introduction to Probability - Chapter 6 doc

... Frequency 1 17 .17 168 1 . 168 1 -2 17 .17 167 8 . 167 8 3 16 . 16 162 6 . 162 6 -4 18 .18 169 6 . 169 6 5 16 . 16 168 6 . 168 6 -6 16 . 16 163 3 . 163 3 Table 6. 1: Frequencies for dice game. = 9 6 − 12 6 = − 3 6 = −.5 . This ... 2 46 CHAPTER 6. EXPECTED VALUE AND VARIANCE God does not exist God exists p 1 −p believe −u v not believe 0 −x Table 6. 4: Payoffs. Age S...

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... the average, you will have to wait. Write a program to see if your conjecture is right. 2.2. CONTINUOUS DENSITY FUNCTIONS 65 -1 -0 .5 0.5 1 1.5 2 0.2 0.4 0 .6 0.8 1 -1 -0 .5 0 0.5 1 1.5 2 0.25 0.5 0.75 1 1.25 1.5 1.75 2 F ... 64 CHAPTER 2. CONTINUOUS PROBABILITY DENSITIES -1 1 2 3 0.2 0.4 0 .6 0.8 1 -1 1 2 3 0.2 0.4 0 .6 0.8 1 F Z (z) f (z) Z Figure 2.15: Dis...

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... 1 56 CHAPTER 4. CONDITIONAL PROBABILITY Y -1 012 X -1 0 1/ 36 1 /6 1/12 0 1/18 0 1/18 0 1 0 1/ 36 1 /6 1/12 2 1/12 0 1/12 1 /6 Table 4 .6: Joint distribution. 36 A die is thrown twice. ... can expect to live to age 60 , while 57. 062 % can expect to live to age 80. Given that a woman is 60 , what is the probability that she lives to age 80? This is an example...

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... 264 6 2 2934 3 3352 4 3000 5 3357 6 2892 7 365 7 8 3025 9 3 362 10 2985 11 3138 12 3043 13 269 0 14 2423 15 25 56 16 24 56 17 2479 18 22 76 19 2304 20 1971 21 2543 22 267 8 23 2729 24 2414 25 261 6 26 ... 161 . 2 16 CHAPTER 5. DISTRIBUTIONS AND DENSITIES Female Male A 37 56 93 B 63 60 123 C 47 43 90 Below C 5 8 13 152 167 319 Table 5.8: Calculus class data. Female Male A 44.3...

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... Chebyshev 100 .31731 1.00000 200 .15730 .50000 300 .083 26 .33333 400 .04550 .25000 500 .02535 .20000 60 0 .01431 . 166 67 700 .00815 .142 86 800 .00 468 .12500 900 .00270 .11111 1000 .00157 .10000 Table ... how 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 0.0...

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... 30 35 40 -1 0 -8 -6 -4 -2 2 4 6 8 10 Figure 12.1: A random walk of length 40. Theorem 12.1 The probability of a return to the origin at time 2m is given by u 2m =  2m m  2 −2m . The probability ... (q/p) a . B’s top counter is valued (q/p) a+1 , and so on downwards until his bottom counter which is valued (q/p) a+b . After each game the loser’s top counter is transfe...

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... DISCRETE PROBABILITIES 3 .203309 . 762 057 .151121 .62 3 868 .932052 .415178 .7 167 19 . 967 412 . 069 664 .67 0982 .352320 .049723 .7502 16 .784810 .089734 . 966 730 .9 467 08 .380 365 .027381 .900794 Table 1.1: ... 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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... 45 10 1 9 1 9 36 84 1 26 1 26 84 36 9 1 8 1 8 28 56 70 56 28 8 1 7 1 7 21 35 35 21 7 1 6 1 6 15 20 15 6 1 5 1 5 10 10 5 1 4 1 4 6 4 1 3 1 3 3 1 2 1 2 1 1 1 1 j = 0 1 2 3 4 5 6 7 8 9 10 Figure ... PERMUTATIONS 79 Number of people Probability that all birthdays are different 10 .8830518 20 .588 561 6 30 .29 368 38 40 .108 768 2 50 .02 962 64 60 .0058773 70 .0008404 80 .0000...

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... m(2)m(1) = 1 6 · 1 6 + 1 6 · 1 6 = 2 36 , P (S 2 =4) = m(1)m(3) + m(2)m(2) + m(3)m(1) = 1 6 · 1 6 + 1 6 · 1 6 + 1 6 · 1 6 = 3 36 . Continuing in this way we would find P (S 2 =5)=4/ 36, P (S 2 =6) =5/ 36, P ... function: m =  1234 56 1 /61 /61 /61 /61 /61 /6  . The distribution function of S 2 is then the convolution of this distribution with itself. Thus, P (S 2...

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... the probability that the sum of the rolls lies between 1400 and 1550? The sum is a random variable S 420 = X 1 + X 2 + ···+ X 420 , where each X j has distribution m X =  1234 56 1 /61 /61 /61 /61 /61 /6  We ... theorem. 5 R. M. Kozelka, “Grade-Point Averages and the Central Limit Theorem,” American Math. Monthly, vol. 86 (Nov 1979), pp. 77 3-7 77. 6 W. Feller, Introduction to...

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