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Topic 6: Convergence and Limit Theorems

Topic 6: Convergence and Limit Theorems Sum of random variables Laws of large numbers Central Limit theorem Convergence of sequences of RVsES150 Harvard SEAS1 Sum of random variablesLetX1,X2, .., Xnbe a sequence of random variables. DefineSnasSn=X1+X2+ +Xn The mean and variance ofSbecomeE[Sn]=E[X1]+E[X2]+ +E[Xn]var(Sn)=n k=1var(Xk)+n j=1j =kn k=1cov(Xj,Xk) IfX1,X2, .., Xnareindependentrandom variables, thenvar(Sn)=n k=1var(Xk)The characteristic function can be used to calculate the joint pdf as Sn( )=E[ej Sn]= X1( ) Xn( )fSn(x)=F 1{ X1( ) Xn( )}ES150 Harvard SEAS2 Sum of a random number of independent RVs Consider the sum RVsXiwith finite mean and varianceSN=N k=1 XkwhereNis a random variable independent of theXk.

Topic 6: Convergence and Limit Theorems ... – This is the Central Limit Theorem (CLT) and is widely used in EE. ES150 – Harvard SEAS 7 • Examples: 1. Suppose that cell-phone call durations are iid RVs with μ = 8 and σ = 2 (minutes). – Estimate the probability of 100 calls taking over 840 minutes.

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