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Math Help - If X1, X2, ...., Xn are iid does that necessarily mean X1^2, X2^2, ..., Xn^2 are iids

  1. #1
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    If X1, X2, ...., Xn are iid does that necessarily mean X1^2, X2^2, ..., Xn^2 are iids

    If X1, X2, Xn are iids, this means that:

    Var(X1+X2+...+Xn) = Var(X1) + Var(X2) +...+Var(Xn), right??

    but does : Var(X1^2 + X2^2 + ... + Xn^2) = Var(X1^2) + Var(X2^2)+...+Var(Xn^2)?
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  2. #2
    Super Member girdav's Avatar
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    Re: If X1, X2, ...., Xn are iid does that necessarily mean X1^2, X2^2, ..., Xn^2 are

    If X_1,\ldots,X_n are independent then so are X_1^2,\ldots,X_n^2. To see that, take a_1\leq b_1,\ldots,a_n\leq b_n and let g:x\mapsto x^2. We have P\left(\bigcap_{j=1}^nX_j^2\in\left[a_j,b_j\right]\right) =P\left(\bigcap_{j=1}^ng(X_j)\in\left[a_j,b_j\right]\right) =P\left(\bigcap_{j=1}^nX_j\in g^{-1}(\left[a_j,b_j\right])\right),
    and since X_1,\ldots,X_n are independent, we have P\left(\bigcap_{j=1}^nX_j\in g^{-1}(\left[a_j,b_j\right])\right)=\prod_{j=1}^nP\left(X_j\in g^{-1}(\left[a_j,b_j\right])\right) and we are done.
    We conclude that your formula is true if Var(X_1^2) exists (but the problem doesn't come from the independence).
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