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Approximation of Gaussian integral arising in population genetics

  1. Jun 4, 2012 #1
    The following problem arises in the context of a paper on population genetics (Kimura 1962, p. 717). I have posted it here because its solution should demand only straightforward applications of tools from analysis and algebra. However, I cannot figure it out.

    1. The problem statement, all variables and given/known data
    Let z = 4 N s [1 - 2 N v x (1 - x)]

    G(x) = e^-∫z dx

    u(p) = [∫G(x)dx (from x = 0 to p)] / [∫G(x)dx (from x = 0 to 1)]

    Assume that v << 1, and 2Nv << 1. Assume further that Ns is large, but (8 N^2 s v) is very small.

    We have u = 2s - v.

    My question is: how is this obtained?

    2. The attempt at a solution
    One first multiplies z out, getting
    z = 4Ns - 8N^2 v s x (1 - x)

    Then one can plug this into the expression for G(x)
    G(x) = e^-∫z dx

    At this point, I have tried to take 8N^2 v s -> 0, and so we are left with the expression
    G(x) = e^-∫4Ns dx
    However, I do not know if it is justified to do this now, and this may be where I go wrong. Nevertheless, if we do this, we can simply integrate ∫4Ns dx = 4Nsx, and so G(x) = e^-4Nsx. Then we plug this expression into the u(p) expression, but this does not yield the correct solution.

    I would be very grateful for any advice.
     
  2. jcsd
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