# Higher Moments of Normal Distribution

# Higher Moments of Normal Distribution

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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 *X*^{n} by taking the *n*th 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

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