Power Spectral Density from Probability Density Function
If the $x[n]$ are i.i.d. random variables with mean $\mu$ and variance $\sigma^2$, then the power spectral density of this discrete-time random process (which is effectively white noise plus a possibly nonzero mean $\mu$) is what you have calculated.  The shape of the common pdf of the random variables is irrelevant in all this except insofar as the shape determines the mean and variance; for example. you would get the same PSD if the $x[n]$ were Gaussian random variables with mean $\mu$ and variance $\sigma^2$.