6.041 / 6.431 20. Central Limit Theorem

n =2 LECTURE 20 THE CENTRAL LIMIT THEOREM • Readings: Section 5.4 • X1, . . . , Xn i.i.d., finite variance !2 • “Standardized” Sn = X1 + ···+ Xn: Zn = Sn − E[Sn] !Sn = Sn − nE[X] "n! – E[Zn]=0, var(Zn)=1 • Let Z be a standard normal r.v. (zero mean, unit variance) • Theorem: For every c: P(Zn # c) $ P(Z # c) • P(Z # c) is the standard normal CDF, !(c), available from the normal tables Usefulness • universal; only means, variances matter • accurate computational shortcut • justification of normal models What exactly does it say? • CDF of Zn converges to normal CDF – not a statement about convergence of PDFs or PMFs Normal approximation • Treat Zn as if normal – also treat Sn as if normal Can we use it when n is “moderate”? • Yes, but no nice theorems to this effect • Symmetry helps a lot 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 15 20 25 30 35 40 0 0.02 0.04 0.06 0.08 0.1 0.12 n = 8 0 10 20 30 40 50 60 70 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 n = 16 30 40 50 60 70 80 90 100 0 0.01 0.02 0.03 0.04 0.05 0.06 n = 32 0 5 10 15 20 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 5 10 15 20 25 30 35 0 0.02 0.04 0.06 0.08 0.1 n =4 100 120 140 160 180 200 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 n =32 0 2 4 6 8 0 0.05 0.1 0.15 0.2 0.25 The pollster’s problem using the CLT • f : fraction of population that “ ...%% • ith (randomly selected) person polled: Xi = !" # 1, if yes, 0, if no. • Mn =(X1 + ···+ Xn)/n • Suppose we want: P(|Mn − f | & .01) # .05 • Event of interest: |Mn − f | & .01 $$$$ X1 + ···+ Xn − nf n $$$$ & .01 $$$$$ X1 + ···+ Xn − nf "n! $$$$$ & .01"n ! P(|Mn − f | & .01) ' P(|Z| & .01"n/!) # P(|Z| & .02"n) Apply to binomial • Fix p, where 0