According to Digital Communication textbook, auto-correlation of a random process denotes the expectation of the random process multiplied by its time-delayed. $$R_X(\tau)=E[X(t)X(t+\tau)]$$ where $$X(t) \mbox{ is a random process}$$
In addition, for a stationary random process $X(t)$, the power spectral density is the Fourier transform of the auto-correlation function $$S_X(f)=\mathbb{F}[R_X(\tau)]$$
The white noise is defined by $$S_N(f)=\frac{N_0}{2}$$
Now, my curiosity was aroused. What is the p.d.f, c.d.f. or m.g.f. with regard to the White noise?
Therefore, I proceeded reversely.
$$R_X(\tau)=\mathbb{F}^{-1}[S_X(f)]=\frac{N_0}{2}\delta(\tau)$$
$$\therefore E[X(t)X(t+\tau)]=\delta(\tau)$$
My progress of finding $X(t)$ is stopped here.
Because I don't know how to solve that equation, I am trying to guess $X(t)$ and then show that the equation is correct or not through substitution.
Can someone give me any hints or good links about this problem?
Maybe, I should've just accept without any curiosities. Thanks to this curiosity, there is little progress of communication. Damn :(