Cental Limit Theorem Department of Mathematics and 中心极限定理的数学系.pptVIP

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2018-06-19 发布于福建
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Cental Limit Theorem Department of Mathematics and 中心极限定理的数学系.ppt

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Cental Limit Theorem Department of Mathematics and 中心极限定理的数学系

Central Limit Theorem Example: (NOTE THAT THE ANSWER IS CORRECTED COMPARED TO NOTES5.PPT) 5 chemists independently synthesize a compound 1 time each. Each reaction should produce 10ml of a substance. Historically, the amount produced by each reaction has been normally distributed with std dev 0.5ml. What’s the probability that less than 49.8mls of the substance are made in total? What’s the probability that the average amount produced is more than 10.1ml? 3. Suppose the average amount produced is more than 11.0ml. Is that a rare event? Why or why not? If more than 11.0ml are made, what might that suggest? Answer: Central limit theorem: If E(Xi)=m and Var(Xi)=s2 for all i (and independent) then: X1+…+Xn ~ N(nm,ns2) (X1+…+Xn)/n ~ N(m,s2/n) Lab: Let Y = total amount made. Y~N(5*10,5*0.52) (by CLT)Pr(Y49.8) = Pr[(Y-50)/1.12 (49.8-50)/1.12]=Pr(Z -0.18) = 0.43 Let W = average amount made.W~N(10,0.52/5) (by CLT)Pr(W 10.1) = Pr[Z (10.1 – 10)/0.22]=Pr(Z 0.45) = 0.33 Lab (continued) One definition of rare:It’s a rare event if Pr(W 11.0) is small(i.e. if “Seeing probability of 11.0 or something more extreme is small”)Pr(W11) = Pr[Z (11-10)/0.22] = Pr(Z4.55) = approximately zero. This suggests that perhaps either the true mean is not 10 or true std dev is not 0.5 (or not normally distributed…) Let Xi = 1 if person i thinks the Presidentis hiding something and 0 otherwise. Suppose E(Xi) = p and Var(Xi) = p(1-p) and each person’s opinion is independent. Let Y = total number of “yesses” = X1+…+ X1006 Y ~ Bin(1006,p) Suppose p = 0.36 (this is the estimate…) What is Pr(Y 352)? Normal Approximation to the binomial CDF Normal Approximation to the binomial CDF Probability meaning of 6 sigma Even if you shift the process mean for the center of the specifications to 1.5 standard deviations toward one of the specifications, then you will expect no more than 3.4 out of a million defects outside of the specification toward which you shifted. (I know it’s convolut