Introduction to Probability phần 10 doc

... same fixed probability vector w. 14 If P is a reversible Markov chain, is it necessarily true that the mean time to go from state i to state j is equal to the mean time to go from state j to state ... WALKS IN EUCLIDEAN SPACE 473 5 10 15 20 25 30 35 40 -10 -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 origi...

Ngày tải lên: 09/08/2014, 23:20

... lines crossed per experiment. Write a pro- gram to simulate this experiment and run your program for the number of experiments equal to 100 , 100 0, and 10, 000. Compare your results with the methods ... .8 1.0 .332 100 00 100 00 100 00 Figure 2 .10: Bertrand’s paradox. length L > √ 3 if its midpoint has distance d < 1/2 from the origin (see Figure 2.9). The following calculat...

Ngày tải lên: 09/08/2014, 23:20

... THEOREM Exercises 1 Let S 100 be the number of heads that turn up in 100 tosses of a fair coin. Use the Central Limit Theorem to estimate (a) P (S 100 ≤ 45). (b) P (45 < S 100 < 55). (c) P (S 100 > 63). (d) ... 102 0. The standard deviation for the number that accept is √ 1700 · .6 · .4 ≈ 20. Thus we want to estimate the probability P (S 1700 > 106 0) = P (S 1700...

Ngày tải lên: 09/08/2014, 23:20

... by a probability distribution on the set of states, which we will call a probability vector. A probability vector with r components is a row vector whose entries are non-negative and sum to 1. ... going to a certain four-year medical school in northern New England has, each year, a probability q of flunking out, a probability r of having to repeat the year, and a probability...

Ngày tải lên: 09/08/2014, 23:20

... with careful planning one would have to be extremely lucky to be able to stop so cleverly. The second author likes to trace his interest in probability theory to the Chicago World’s Fair of 1933 ... referring to a continuous random variable X (say with a uniform density function), it is customary to say that “X is uniformly distributed on the interval [a, b].” It is also custo...

Ngày tải lên: 04/07/2014, 10:20

... CHAPTER 4. CONDITIONAL PROBABILITY Number having The results Disease this disease ++ +– –+ –– d 1 3215 2 110 301 704 100 d 2 2125 396 132 1187 410 d 3 4660 510 3568 73 509 Total 100 00 Table 4.3: Diseases ... particular disease. What the doctor wants to know is the posterior probability for the particular disease, given the outcomes of the tests. Example 4.16 A doctor is trying t...

Ngày tải lên: 04/07/2014, 10:20

... result of 100 Bernoulli trials with p =1 /100 0. The expected value of S 100 is λ = 100 (1 /100 0) = .1. The exact probability that S 100 = j is b (100 , 1 /100 0,j), and the Poisson approximation is e −.1 (.1) j j! . In ... distribution to X? If it is appropriate, what probability should we assign to each outcome? (a) Take the first 100 students who enter the cafeteria to eat...

Ngày tải lên: 04/07/2014, 10:20

... 61 62 63 64 1 /10 2 /10 4 /10 2 /10 1 /10  . (a) Find E(F ) and V (F ). (b) Define T = F − 62. Find E(T ) and V (T ), and compare these answers with those in part (a). (c) It is decided to report the ... favorable no matter how much you pay to play it? (b) Assume that you only receive 2 10 dollars if any number greater than or equal to ten tosses are required to obtain the first...

Ngày tải lên: 04/07/2014, 10:20

... ≤ .21 n(.1) 2 = 21 n . Thus, if n = 100 , P (|A 100 − .3|≥.1) ≤ .21 , or if n = 100 0, P (|A 100 0 − .3|≥.1) ≤ .021 . These can be rewritten as P (.2 <A 100 <.4) ≥ .79 , P (.2 <A 100 0 <.4) ≥ .979 . These ... so to speak, a certain fate. I do now know whether Plato wished to aim at this in his doctrine of the universal return of things, according to which he predicte...

Ngày tải lên: 04/07/2014, 10:20

... -50 000000000000 -100 111111249797300 233420000000 0 100 000000000 -50 2100 00000000 0 3347111714111 1101 6251300 000000000000 -100 12211 3100 000 -50 000000000000 -100 2 3100 0000000 0 3100 00000000 50 100 000000000 -50 344 7101 191112141 310 550 13349579886 3100 104 66 9101 30000 ... follows: 365 392 CHAPTER 10. GENERATING FUNCTIONS Z 1 Z 2 Z 3 Z 4 Z 5 Z 6 Z 7 Z 8 Z 9 Z 10 Z 1...

Ngày tải lên: 04/07/2014, 10:20