If with and , then and .

For the mean, write the integral as an odd part plus a constant part:

The first integrand is odd, so its integral vanishes, and the second integral equals ; hence . For the variance, substituting gives

If and with , then

Indeed, for we have and , so the change-of-variables formula gives

If , then . Writing for the distribution function of the standard normal, its density is , and since while ,

Existing tables of therefore give for any normal random variable; for instance

If and are independent, their moment generating functions multiply:

so : sums of independent normal random variables are normal.

Equivalently to the density definition, a Gaussian random variable is any random variable of the form

When this has density ; allowing also admits the degenerate case .