Results 1 to 4 of 4

Math Help - Proof for mean of non-negative discrete random variable

  1. #1
    Newbie
    Joined
    Mar 2009
    Posts
    6

    Proof for mean of non-negative discrete random variable

    For a nonnegative integer-valued discrete random variable X, show that
    E(X)=x=0 (to infinity) P(X>x). Can anyone help me solve this please?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jul 2008
    Posts
    138
    So

    \sum_{x=0}^\infty P(X>x) = \sum_{x=0}^\infty \sum_{y=x+1}^\infty P(X=y)

    This is a triangular region. So we can switch the order of the sum if we do it carefully.

     = \sum_{y=1}^\infty \sum_{x=0}^{y-1} P(X=y)

     = \sum_{y=1}^\infty y P(X=y)

    Can you finish it?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2009
    Posts
    6
    Thank so much for your reply!
    So to finish it then:

    =x=0 (to infinity)xP(x)=E(x)?
    Because x=0 --> y=1?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Jul 2008
    Posts
    138
    Well, there are no x's anymore. The x and y are just the summing variables.

    The main observation at this point is that the sum:
    \sum_{y=1}^\infty yP(X=y)
    almost is E[X] (whether it is x or y doesn't matter, it is just a summing variable). The only thing missing is the 0 term. The sum should go from 0 to \infty, whereas this one starts at 1. But the 0 term is
    0P(X=0)=0.

    So \sum_{y=1}^\infty yP(X=y)  = 0P(X=0)+\sum_{y=1}^\infty yP(X=y) = \sum_{y=0}^\infty yP(X=y) =E[X]
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Discrete Random Variable Help
    Posted in the Statistics Forum
    Replies: 2
    Last Post: November 20th 2010, 12:11 PM
  2. Replies: 3
    Last Post: January 13th 2010, 10:44 AM
  3. Discrete random variable
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: April 15th 2008, 08:06 AM
  4. discrete random variable
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: October 25th 2007, 09:20 AM
  5. Discrete Random Variable
    Posted in the Statistics Forum
    Replies: 2
    Last Post: September 23rd 2007, 08:40 AM

Search Tags


/mathhelpforum @mathhelpforum