Definitions of Gaussian, mean, variance

unnormalized Gaussian

G[x_] = Exp[-(x - m)^2/(2 s ^2)]

Integrate[G[x],{x,-∞,∞},Assumptions->s > 0]

Normalize it

G[x_] = 1/(Sqrt[2 Pi] s) Exp[-(x - m)^2/(2 s ^2)]

Integrate[G[x],{x,-∞,∞},Assumptions->s > 0]

Compute first and second moments and verify that
< x^2 > - < x > ^2 = < (x - < x >)^2 >

Integrate[x G[x],{x,-∞,∞},Assumptions->s > 0]

Integrate[x^2 G[x], {x,-∞,∞},Assumptions->s > 0]

Integrate[(x-m)^2 G[x], {x,-∞,∞},Assumptions->s > 0]

s^2 is called the variance.  s is called the standard deviation.

For a Gaussian, all higher moments are determined by m and s.  Hence, m and s serve to completely specifiy the distribution

Integrate[x^3 G[x], {x,-∞,∞},Assumptions->s > 0]
Integrate[x^4 G[x], {x,-∞,∞},Assumptions->s > 0]
Integrate[x^5 G[x], {x,-∞,∞},Assumptions->s > 0]