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1、Limit TheoremLimit Theorem 极限定理极限定理 Probability and Mathematical Statistics (概率与数理统计) Xi ZHANG In many cases, we dont need to calculate exactly the probability but roughly know it Especially when the probability is very large or very small e.g Phaze tomorrow in Lhasa = ? For example, suppose tossing
2、 a coin 1000 times, we would like to know if the probability of consecutive 17 appearance of heads Let be the number of occurrences of 17 consecutive heads in 1000 coin flips. N = I1+ + I984 EIi = P(Ii= 1) = 1/217 EN = 984 1/217=0.007507 Outlines Chebyshevs Inequality and the Weak Law of Large Numbe
3、rs (切比雪夫不 等式及弱大数定律） The Central Limit Theorem(中心极限定理) The Strong Law of Large Numbers（强大数定律） Summary Markovs Inequality （马尔可夫不等式） Proposition: Markovs Inequality If is a random variable that takes only nonnegative values, then, for any value 0 E X P Xa a Hence, PN 1 EN / 1 0.75%. EX = EX | X a P(X a
4、) + EX | X 4 The Weak Law of Large Numbers （弱大数定理） Proposition: If () = 0, then 1P XE X Theorem: The weak law of large numbers Let 1,2,. be a sequence of independent and identically dist ributed random variables, each having finite mean = . Then, for any 0, 1 0 as n XX Pn n The Central Limit Theorem
5、 （中心极限定理） Let 1,2,. be a sequence of independent and identically distributed random variables, each having mean and variance 2. Then the distribution of tends to the standard normal as . That is, for , 1n XXn n 2 /2 1 1 as 2 a x n XXn Paedxn n Example An instructor has 50 exams that will be graded i
6、n sequence. The times required to grade the 50 exams are independent, with a common distribution that has mean 20 minutes and standard deviation 4 minutes. Approximate the probability that the instructor will grade at least 25 of the exams in the first 450 minutes of work. = =1 25 = =1 25 = 25 20 =
7、500 = =1 25 = 25 16 = 400 450 = 500 400 450 500 400 2.5 = 1 2.5 = 0.006 Exercise If 10 fair dice are rolled, find the approximate probability that the sum obtained is between 30 and 40, inclusive. The Strong Law of Large Numbers Theorem: The strong law of large numbers Let 1,2,. be a sequence of ind
8、ependent and identically distributed ra ndom variables, each having a finite mean = . Then, with proba bility 1, 12 as n XXX n n Comparison between weak law of large numbers Weak law of large numbers: For any specified large value , 1+ is likely to be near , it does not say that (1+ +)/ is bound to
9、stay near for all values of larger than . Thus, it leaves open the possibility that large valu es of |(1+ +)/ | can occur infinitely often (though at infrequen t intervals). The strong law shows that this cannot occur Summary Chebyshevs Equality Weak Law of Large Numbers The Central Limit Theorem Th
10、e Strong Law of Large Numbers 2 2 P Xk k 1 0 as n XX Pn n 2/2 1 1 as 2 a x n XXn Paedxn n 12 as n XXX n n Homework 3 will be due on Nov. 6th Midterm exam will be on Nov. 17th Review lesson and sample questions will be given on Nov. 13th. Markov chainsstochastic processes Stochastic processes Many re
11、al-world systems contain uncertainty and evolve over time. Stochastic processes (and Markov chains) are probability models for such systems. A discrete-time stochastic process is a sequence of random variables: 0,1,2, . . . typically denoted by Time: = 0,1,2,. State: v-dimensional vector, = (1,2,.,)
12、 In general, there are states (a finite # of states): 1,2,., Random walk problem A stochastic process whose state space is given by the integer = 0,1,2, is said to be a random walk if, for some number 0 1, ,+1= = 1 ,1, = 0,1,2, A Markov Chain Definition of the Markov chain A stochastic process is ca
13、lled a Markov chain if +1= = ,1= 1,1= 1,0= 0 = +1= = = transition probabilities 转移概率 Discrete time means = 0,1,2,. The future behavior of the system depends only on the current state and not on any of the previous states For all states, = 1 0 , 000 0 transition matrix 转移概率矩阵 X1X2Xn-1Xn = ,1= 1,1= 1,
14、0= 0= 1, 1= 1,1= 1,0= 0 = 1,2,11,20,10= 0 Matrix representation Each directed edge AB is associated with the positive transition probability from A to B 0100 0.800.20 0.300.50.2 00.0500.95 AB B A C C D D 0.2 0.3 0.5 0.05 0.95 0.2 0.8 1 Example_Gamblers Ruin At time zero the gambler has 0= $2, and each day he makes a $1 bet.
