Determining the parameters of a uniform distribution from its autocorrelation function

Question

I have a noise process v(t) which is wide sense stationary (WSS) and uniformly distributed with autocorrelation $R_{vv}(\tau)=c^2e^{-\beta\mid\tau\mid}$ where c =0.1.

How would I find $E[v(t)]$, the expected value of v(t)?

I think the way to solve is to take the Fouier transform ($FT$) of $R_{vv}$ and evaluate the function at 0, the DC value. I.e. $S_{vv}(\omega)=FT\{R_{vv}\}$, $E[v(t)] = S_{vv}(0)$

Next, I want to determine the amplitude range of v(t). I believe this is equivalent to finding the bounds of the uniform distribution. I am unsure how to proceed with finding this. Here is what I have tried:

1. I would think the autocorrelation function would help because $R_{vv}=E[v(t)v(t+\tau)]$ but I don't see how to derive PDF properties from this.

2. The power spectral density (PSD) contains the same information as $R_{vv}$ so if 1 can't help me, this can't either.

Any ideas?

2.