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CLT Concept Question

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  • CLT Concept Question

    i know you can approximate

    X1+X2+X3+...+Xn by N(nu, n*variance)

    also Var(nX) = n^2 * Var(x)

    but isnt this kind of contradictory?

    i mean the CLT says n*Var and the second line says n^2 * Var

    im kinda confused

  • #2
    you are thinking of this wrong.

    The central limit theorem only applies to a random sample of independent, identical distributions. So, X1+X2....Xn=nX. Each X has the same variance, so adding up n independent identical variances is equivalent to n*var(X). It follows that Var(X1+X2...Xn)=Var(x1)+Var(X2)+...+Var(Xn)=nVar(X ). And what do we know? We know they all have the same variance (identical) and are all independent(Cov(Xi,Xj)=0).

    Var(aX)=a^2*Var(X) is a transformation. Y=aX Var(Y)=Var(aX)=a^2Var(x). There is no random sample here.

    Let me know if this makes sense...If you have ASM manual, I recommend reading his explanation of CLT a few times and be very precise and detailed about it, noting that a couple of those words (identical, independent) carry a lot of weight
    dystrother
    Actuary.com - Level III Poster
    Last edited by dystrother; July 15 2008, 09:14 PM.

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