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Math Help - Are the sums of independent RVs necessarily independent of each other?

  1. #1
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    Are the sums of independent RVs necessarily independent of each other?

    Suppose that X_1,X_2, \ldots , X_n are i.d.d. to some distributions  ( \mu _X , \sigma ^2 _X ) and  Y_1, \ldots , Y_m are i.d.d. to  ( \mu _Y , \sigma _Y ^2 ) .

    If all the X's and Y's are indepedent to one aother, does that also mean that the sums  X_1 + \ldots + X_n is also independent to  Y_1 + \ldots + Y_m ?

    So I'm trying to prove it by doing it striaght from the definition:

    Pr[ X_1 + \dots + X_n \cap Y_1 + \dots + Y_m ] and try to get it to become  Pr[X_1 + \dots + X_n ] \cdot Pr[Y_1 + \dots + Y_m ]

    But I'm getting stuck on the left hand side, not sure how to break it down...

    Any help, please? Thank you!
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  2. #2
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    Re: Are the sums of independent RVs necessarily independent of each other?

    Hey tttcomrader.

    You need further information to answer your question. For example if Yn = Xn^2 then clearly you don't have independence.
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