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Show that if X is a bounded random variable, then E(X) exists.

  1. Jan 24, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that if X is a bounded random variable, then E(X) exists.


    2. Relevant equations



    3. The attempt at a solution

    I am having trouble of finding out where to begin this proof.


    This is what I got so far:

    Suppose X is bounded. Then there exists two numbers a and b such that P(X > b) = 0 and P(X < a) = 0 and P(a <= X <= b) = 1.



    I have no idea if I am even doing this right. Anyone wanting to take a crack at this one? Thanks.
     
    Last edited: Jan 24, 2012
  2. jcsd
  3. Jan 25, 2012 #2

    Stephen Tashi

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    How do your course materials define a "bounded random variable"? The claim "if X is a bounded random variable, then E(X) exists" isn't true using the usual definition of a "bounded random variable".

    Perhaps your problem assumes the domain of X is bounded as well as the range.
     
  4. Jan 25, 2012 #3
    The book specifically defined X as bounded as the following:

    |X| < M < ∞ .


    Here is the whole question, word for word:

    Show that if a random variable is bounded—that is, |X| < M < ∞—then
    E(X) exists.


    I do not know about the range though.
     
    Last edited: Jan 25, 2012
  5. Jan 25, 2012 #4

    Stephen Tashi

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    I see what the book is doing now. I was thinking of a probability density function being "bounded" as meaning it is a function with a bounded range on a possibly infinite domain. Your book means the domain is bounded and you probably get to assume the range of the function is [0,1] or a subset of it. So you can bound the integral of the expression x f(x) by (M)(1) = M.

    Have you studied theorems that say something to the effect that bounded (in range) continuous function on a closed interval is integrable?
     
  6. Jan 25, 2012 #5

    lanedance

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    Homework Helper

    start witha a well-behaved continously distributed random variable (no delta functions)

    then start with the definition of E(X) in integral form.

    You know integral of the distribution function p(x) converges, so can you show the integrand in the expectation is always less than something you know converges eg. xp(x) < c, or xp(x) < cp(x) for all x?
     
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