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John Cook is an applied mathematician working in Houston, Texas. His career has been a blend of research, software development, consulting, and management. John is a DZone MVB and is not an employee of DZone and has posted 170 posts at DZone. You can read more from them at their website. View Full User Profile

Higher Moments of Normal Distribution

11.05.2012
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Sometimes a little bit of Python beats a Google search.

Last week I needed to look up the moments of a normal distribution. The first two moments are common knowledge, the next two are easy to find, but I wasn’t able to find the higher moments.

Here is a little Sage code that produces a table of moments for the normal distribution. (Sage is a Python-based mathematical computing environment.) The code computes the expected value of Xn by taking the nth derivative of the moment generating function and setting its argument to zero.

var('m, s, t')
mgf(t) = exp(m*t + t^2*s^2/2)
for i in range(1, 11):
    derivative(mgf, t, i).subs(t=0)

Here's the output:

m
m^2 + s^2
m^3 + 3*m*s^2
m^4 + 6*m^2*s^2 + 3*s^4
m^5 + 10*m^3*s^2 + 15*m*s^4
m^6 + 15*m^4*s^2 + 45*m^2*s^4 + 15*s^6
m^7 + 21*m^5*s^2 + 105*m^3*s^4 + 105*m*s^6
m^8 + 28*m^6*s^2 + 210*m^4*s^4 + 420*m^2*s^6 + 105*s^8
m^9 + 36*m^7*s^2 + 378*m^5*s^4 + 1260*m^3*s^6 + 945*m*s^8
m^10 + 45*m^8*s^2 + 630*m^6*s^4 + 3150*m^4*s^6 + 4725*m^2*s^8 + 945*s^10

Published at DZone with permission of John Cook, author and DZone MVB. (source)

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