Results 1 to 2 of 2

Math Help - Evaluate series np(1-p)^k

  1. #1
    Junior Member
    Joined
    May 2007
    Posts
    30

    Evaluate series np(1-p)^k

    Trying to derive the expectation for the geometric distribution. Can someone help me derive the identity of the series n(p)^(n-1) = 1/(1-p)^2?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Aug 2007
    From
    USA
    Posts
    3,111
    Thanks
    2
    Quote Originally Posted by CrazyAsian View Post
    Trying to derive the expectation for the geometric distribution. Can someone help me derive the identity of the series n(p)^(n-1) = 1/(1-p)^2?
    I'm not convinced you have written what you want. Anyway, it's called "Geometric" for a reason. Everything you ever leared about Geometric Series applies, here.

    Remember this?

    \frac{a}{1-r} = a + ar + ar^{2} + ar^{3} + ...

    You ALMOST have that. You're only additional concern is that pesky linearly increasing coefficient. It is easily removed by an old trick.

    If p + 2pq + 3pq^{2} + 4pq^{3} + ... = S

    Then pq + 2pq^{2} + 3pq^{3} + 4pq^{4} + ... = Sq

    Subtracting the second from the first gives:

    p + pq + pq^{2} + pq^{3} + ... = S - Sq = S(1-q)

    Are we getting anywhere? Let's work with the left hand side - WAIT! That's just our friend the Geometric Series. We know its sum.

    \frac{p}{1-q} = S(1-q)

    Solve for S, substitute p = 1-q, and I think you're done.

    Note: Before solving for S and before finding the sum of the left hand side, one MAY notice that these are simply the probabiltiies of the possible outcomes of the Geometric Distribution. The sum of those had better be one (1).
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. using a series to evaluate a limit:
    Posted in the Calculus Forum
    Replies: 12
    Last Post: May 7th 2009, 11:44 AM
  2. Evaluate the Series
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 8th 2008, 03:51 PM
  3. Evaluate the series
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 15th 2008, 09:38 AM
  4. Using series to evaluate integrals
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 24th 2007, 09:13 PM
  5. Complex analysis to evaluate series
    Posted in the Calculus Forum
    Replies: 3
    Last Post: November 29th 2006, 08:09 AM

Search Tags


/mathhelpforum @mathhelpforum