Introduction to Probability - Chapter 5 docx

... 2646 2 2934 3 3 352 4 3000 5 3 357 6 2892 7 3 657 8 30 25 9 3362 10 29 85 11 3138 12 3043 13 2690 14 2423 15 255 6 16 2 456 17 2479 18 2276 19 2304 20 1971 21 254 3 22 2678 23 2729 24 2414 25 2616 26 2426 ... CHAPTER 5. DISTRIBUTIONS AND DENSITIES Dark Eyes Light Eyes Dark Hair 28 15 43 Light Hair 9 23 32 37 38 75 Table 5. 6: Observed data. 0 10 20 30 40 0 50 0 1000 150 0 2000...

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... 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 (z) Z f (z) Z Figure 2.17: Distribution and ... PROBABILITIES 51 Length of Number of Number of Estimate Experimenter needle casts crossings for π Wolf, 1 850 .8 50 00 253 2 3. 159 6 Smith, 1 855 .6 32...

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... Chebyshev 100 .31731 1.00000 200 . 157 30 .50 000 300 .08326 .33333 400 .0 455 0 . 250 00 50 0 .0 253 5 .20000 600 .01431 .16667 700 .008 15 .14286 800 .00468 .1 250 0 900 .00270 .11111 1000 .00 157 .10000 Table 8.1: Chebyshev ... 0.6 0.8 1 0 0. 05 0.1 0. 15 0.2 0. 25 0 0.2 0.4 0.6 0.8 1 0 0.0 25 0. 05 0.0 75 0.1 0.1 25 0. 15 0.1 75 n=10 n=20 n=40 n=30 n=60 n=100 Figure 8.1: Bernou...

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... PROBABILITIES 5 5 10 15 20 25 30 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 that ... .51 3. Thus, it is more appropriate to assign a distribution function which assigns probability .51 3 to the outcome boy and probability .487 to the outcome girl...

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... 210 252 210 120 45 10 1 9 1 9 36 84 126 126 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 ... distributions. 78 CHAPTER 3. COMBINATORICS Number of people Probability that all birthdays are different 20 .58 856 16 21 .55 63117 22 .52 43047 23 .4927028 24 .461 655 7 25 .4...

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... Smith Unconditional probability b bb b g g g b 1/4 1/8 1/8 1/8 1/8 1/4 1/4 1/4 1/4 1/4 1 1/2 1/2 1/2 1/2 1 bg gb gg b bb b b 1/4 1/8 1/8 1/4 1/4 1/4 1 1/2 1/2 bg gb Conditional probability 1/2 1/4 1/4 Figure 4.13: Tree for Example 4.26. 148 CHAPTER 4. CONDITIONAL PROBABILITY .001 can not .01 . 95 . 05 + - .001 0 . 05 .949 + - . 051 .949 + - .981 1 0 can not .001 . 05...

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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/ 396 30/396 25/ 396 30/396 2/ 45 3/ 45 1/36 2/36 Figure ... E(Y )=2. 6.2. VARIANCE OF DISCRETE RANDOM VARIABLES 261 1 2 3 4 5 6 0 0.1 0.2 0.3 0.4 0 .5 0.6 2 2 .5 3 3 .5 4 4 .5 5 0 0 .5 1...

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... 298 CHAPTER 7. SUMS OF RANDOM VARIABLES 5 10 15 20 0.0 25 0. 05 0.0 75 0.1 0.1 25 0. 15 Figure 7.4: Chi-squared density with 5 degrees of freedom. 0 5 10 15 20 25 30 0 0.0 25 0. 05 0.0 75 0.1 0.1 25 0. 15 1000 ... (S 3 =3) = P(S 2 =2)P (X 3 =1) 300 CHAPTER 7. SUMS OF RANDOM VARIABLES -1 5 -1 0 -5 5 10 15 0.0 25 0. 05 0.0 75 0.1 0.1 25 0. 15 0.1 75...

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... .4032 2.3 .4893 3.3 .49 95 .4 . 155 4 1.4 .4192 2.4 .4918 3.4 .4997 .5 .19 15 1 .5 .4332 2 .5 .4938 3 .5 .4998 .6 .2 257 1.6 .4 452 2.6 .4 953 3.6 .4998 .7 . 258 0 1.7 . 455 4 2.7 .49 65 3.7 .4999 .8 .2881 1.8 ... 1/2 =50 and √ npq =  100 ·1/2 · 1/2 =5. Thus x 55 = (55 50 ) /5= 1and P (S 100 = 55 ) ∼ φ(1) 5 = 1 5  1 √ 2π e −1/2  = .0484 . To four decimal places, the actual...

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... PROCESSES 383 Generation Probability of dying out 1.2 2 .312 3 .3 852 03 4 .437116 5 .4 758 79 6 .50 5878 7 .52 9713 8 .54 90 35 9 .56 4949 10 .57 82 25 11 .58 9416 12 .59 8931 Table 10.1: Probability of dying ... 0 100000000000 -5 0 210000000000 0 334711171411111016 251 300 00000000000 0-1 00 122113100000 -5 0 00000000000 0-1 00 231000000000 0 310000000000 50 100000000000...

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