Power of AWGN signal

Sometimes I find myself confused over why power of Additive White Gaussian Noise (AWGN) is taken as its variance. Following is a short reasoning why it is so.

We define the mean squared value of a signal x(t), also called power of the signal,  as $P_x = \bar{x^2(t)} = \frac{1}{T} \int_{-T/2}^{T/2} x^2(t) dt$

Now the mean squared values of the three signals can be written as

If we have the frequency domain representation of the signal x(t) as S(f) then Parsevals’s Theorem allows us to calculate the power using S(f) as $P_x = \int_{-\infty}^{\infty} S^2 (f) df$

Ergodicity means that a process has same properties over time as over population of processes of the same kind at any given instant of time. AWGN is ergodic process. It follows a Gaussian distribution with zero mean and variance $\sigma^2$

The variance of a signal can be represented by $\sigma^2_x = E[x^2] - ( E[x] )^2$

Using ergodicity property of AWGN $\sigma^2_x = \bar{ x^2(t) } - \bar{ x(t) }^2$

Because AWGN is zero mean $\bar{ x^2(t) } = \sigma^2_x$