Results 1 to 4 of 4

Math Help - Expected value

  1. #1
    Newbie
    Joined
    Sep 2010
    Posts
    11

    Expected value

    Let X be a discrete random variable. Show that the expected value E(X) minimises the expected sum of squared distances,

    i.e. show that for all a R


     E( (X-a)^2) >= Var(X)

    with quality  a = E(X)


    The question also gives a hint that you can write  (X-a)^2 as ( (X- \mu)+ (\mu-a))^2

    Once I expanded this im unsure what to do.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    (X-a)^2=[(X-\mu)+(\mu-a)]^2=(X-\mu)^2+2(X-\mu)(\mu-a)+(\mu-a)^2

    NOW take expectations and observe that (\mu-a) is a constant and that E(X-\mu)=0

    SO E(X-a)^2=V(X)+(\mu-a)^2\ge V(X)

    and equality occurs when a=MOO
    Last edited by matheagle; October 28th 2010 at 02:55 PM. Reason: typo
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2010
    Posts
    11
    Thanks
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor matheagle's Avatar
    Joined
    Feb 2009
    Posts
    2,763
    Thanks
    5
    and there's no need for the rvs to be discrete at all here
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Expected Value
    Posted in the Statistics Forum
    Replies: 1
    Last Post: April 25th 2011, 03:51 PM
  2. Expected value/ st dev/var
    Posted in the Advanced Statistics Forum
    Replies: 3
    Last Post: September 30th 2010, 02:55 AM
  3. Expected Value
    Posted in the Advanced Statistics Forum
    Replies: 9
    Last Post: September 19th 2010, 11:52 PM
  4. Expected Value
    Posted in the Statistics Forum
    Replies: 3
    Last Post: April 28th 2009, 04:05 PM
  5. expected value
    Posted in the Advanced Statistics Forum
    Replies: 1
    Last Post: January 16th 2009, 08:51 AM

Search Tags


/mathhelpforum @mathhelpforum