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Math Help - Let X,Y be two discrete random variables having finite moments

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    Let X,Y be two discrete random variables having finite moments

    Let X,Y be two discrete random variables having finite moments (I assume this means E(X)<\infty). Show that

    a) if X\geq 0, then E(X)\geq 0

    b) if X\leq Y, then E(X)\leq E(Y)
    HINT: You may consider the variable Y-X.

    Again, with so much to do in so little time, I need help.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Runty View Post
    Let X,Y be two discrete random variables having finite moments (I assume this means E(X)<\infty).
    It means something rather stronger than that, in fact far stronger than is needed here, all you need is that they have finite first moments.

    Show that

    a) if X\geq 0, then E(X)\geq 0
    The detail of thia depends on the type of course you are doing, but in essense this depends on the definite integral of a positive function being positive.


    b) if X\leq Y, then E(X)\leq E(Y)
    HINT: You may consider the variable Y-X.

    Again, with so much to do in so little time, I need help.
    Part b) follows from part a) by introducing the RV Z=X-Y, and the linearity of the expectation operator.


    CB
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    Quote Originally Posted by CaptainBlack View Post
    It means something rather stronger than that, in fact far stronger than is needed here, all you need is that they have finite first moments.

    The detail of thia depends on the type of course you are doing, but in essense this depends on the definite integral of a positive function being positive.


    Part b) follows from part a) by introducing the RV Z=X-Y, and the linearity of the expectation operator.


    CB
    This question is for a statistics course, so integrals are not a part of the material. Also, my Prof. didn't really give us a definition of "finite moments", so I was guessing.
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    MHF Contributor matheagle's Avatar
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    These are discrete, you have all the realization of X nonnegative and since probabilities are also nonnegative...

    E(X)=\sum_{x}xP(X=x)\ge 0 since xP(X=x)\ge 0

    you are adding the product of two terms that cannot be negative.
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    Grand Panjandrum
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    Quote Originally Posted by Runty View Post
    This question is for a statistics course, so integrals are not a part of the material. Also, my Prof. didn't really give us a definition of "finite moments", so I was guessing.
    You can't do statistics without integration (rather you cannot do statistics using only discrete probability distributions other than in a half arsed manner), what I was trying to suggest is that the proof would be slightly more subtle if you were doing a course using measure theory.

    CB
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    Quote Originally Posted by CaptainBlack View Post
    You can't do statistics without integration (you cannot do statistics using only discrete probability distributions), what I was trying to suggest is that the proof would be slightly more subtle if you were doing a course using measure theory.

    CB
    I see. We haven't used integration in our statistics course so far, so that's why I didn't think it was relevant to the question.
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