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Derivation of Lognormal mean

  1. Feb 9, 2009 #1
    Hi, I am curious as to the strategy for integrating the lognormal function to calculate the mean.

    The integral to be solved is:

    [tex]\frac{1}{S\sqrt{2\pi}}\int_{0}^{\infty} \frac{e^{(lnx-M)^{2}}}{2S^{2}} dx [/tex]

    I was trying to do it by a substitution



    to give

    [tex]\frac{1}{S\sqrt{2\pi}}\int_{-\infty}^{\infty} \frac{e^{(y-M)^{2}}}{2S^{2}}e^{y} dy [/tex]

    and then integration by parts, but I keep going round in circles with vdu and what not…

    Can anyone enlighten me on the trick to this?
    Last edited: Feb 9, 2009
  2. jcsd
  3. Feb 9, 2009 #2
    I made this a bit more readable, any ideas?
  4. Feb 10, 2009 #3


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    It looks like you have two errors in your statement. The 2S^2 probably belongs as a divisor in the exponent. Also there should be a minus in the exponent, otherwise the integrand will blow up at both ends.
    In any case the exponent in y is a quadratic polynomial. Recast it in the form -[(y-k)^2]/[2S^2] +n, where k and n are constants. The final answer will be e^n.
  5. Feb 10, 2009 #4
    Thanks, you are right, I have fixed the typos.
    Not sure I understand your suggestion though...

    Do you mean combine the two exponents into one and re-factorise the resultant polynomial?

    The reason I am trying to understand this is that http://mathworld.wolfram.com/LogNormalDistribution.html" [Broken] they state (in reference to the moments of the lognormal distribution) that the following can be found by direct integration:

    [tex]\frac{1}{S\sqrt{2\pi}}\int_{-\infty}^{\infty} e^{-\frac{(y-M)^{2}}{2S^{2}}}e^{y}\; dy \;=\;e^{\frac{M+S^2}{2}}[/tex]
    Last edited by a moderator: May 4, 2017
  6. Feb 11, 2009 #5


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    Now that you've got the exponent right, work out what the constants a and b are for:
    -(y-M)2/2S2 + y =-(y-a)2/2S2 + b.

    I presume that b will turn out to be (M+S2)/2.
  7. Feb 12, 2009 #6
    Thanks! Very much appreciated: you assume correctly!

    Thanks for you help.
  8. Feb 12, 2009 #7


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    but the mean of a lognormal distribution is [itex]e^{\mu + \sigma^2/2}[/itex] not [itex]e^{(\mu + \sigma^2)/2}[/itex]

    it can be derived as a limit of geometric brownian motion
  9. Feb 12, 2009 #8
    yeh, sorry, that is the result I got, neglected to fix the typo...
  10. Feb 13, 2009 #9


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    I decided to work it out myself. I agree with BWV.
  11. Feb 14, 2009 #10
    Yes, sorry, that is what I meant in my last post. I got [itex] e^{\mu + \sigma^2/2}
    [/itex] as well. It was just a typo in my previous posts.
  12. May 15, 2009 #11
    pdf f(x)=1/(x*S*sqrt(2*pie))*integral exp(-((lnx)-m)^2/(2*S^2))dx
    mean=integral((f(x).x.)dx /*x will cancel up in this case)
    mean= integral(1/(S*sqrt(2*pie)*exp(-((lnx)-m)^2/(2*S^2))dx
    solving as you discuseed by assuming ln(x)=y we get
    integral(1/(S*sqrt(2*pie)*exp(-(y-m)^2/(2*S^2))* exp^ydx
    1/(S*sqrt(2*pie))*integral exp(-(y-m)/(2*S^2) + y) dy /* limit - infinity to infity will not change due to replcement of lnx by y)
    1/(S*sqrt(2*pie))*integral exp(-1/(2*s)*(( y-(m+s))^2 -(s^2+2ms))
    exp(1/(S*sqrt(2*pie)*(s^2+2ms)* integral (1/(S*sqrt(2*pie))*(-1/(2*s)*( y-(m+s))^2)

    integral (1/(S*sqrt(2*pie)*(-1/(2*s)*( y-(m+s))^2) is standard normal distrbution with mean (m+s) and variance s. so it will be equal to one.
    one firstpart will be left
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