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1Chapter 11 - The Central Limit Theorem State the central limit theory State the proof of this theoryLet X1, X2, ..., Xn be iid random variables with common mean mu and variance sigma^2. Define Z=[Xbar - mu]/[sigma/sqrt(n)]. Then a n tends to infinity the distribution of Z tends to N(0,1)
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2Chapter 11 - Implications of the Central Limit Theorem The central limit theory implies what 2 things? The rate of convergence towards to the Normal distribution depends on what? If we approximating a discrete random variable (i.e only takes integer values) then we need to use a continuity correction and do what?1. Xbar~N(mu,sigma^2/n)
2. SIGMAXi~N(nmu,nsigma^2)
for large n whatever the distribution of Xi
The symmetry or assymetry of the distribution of X
Replace Pr(X>k) by Pr(X>(k-0.5)) or Pr(X20) with discrete random variable such as Poisson, would become Pr(W>21) - Pr(W>21-0.5)=Pr(W>20.5)
3Chapter 11 - Markov's Inequality, Chebychev's Inequality, WLLN State Markov's Inequality State Chebychev's Inequality State the Weak Law of Large Numbers Inequality State the proofs for each of these inequalitiesFor any positive valued random variable X and any a>0, Pr(X>a)0, Pr(¦X-E(X)¦>a)0, Pr(¦Xbarn-E(X)¦>a)