Power Spectral Density From Variance
The power spectral density is NOT the Fourier transform of the variance. For wide-sense-stationary (WSS) signals, the power spectral density is the Fourier transform of the Autocorrelation function, and the variance is the autocorrelation at time offset $\tau = 0$ (when the WSS process is zero-mean).
Consider for example a white noise process (which is done if the OP's randomly generated samples were completely uncorrelated from sample to sample), to a tightly filtered process, both can have the same variance (which is the power if WSS and zero-mean), but VERY different spectrums. A white noise spectrum is constant over all frequencies while the filtered process would be limited to a narrow bandwidth. The relationship between the two is given by Parseval's theorem; the sum of the squares of each frequency component will be the same in both cases.
See this answer from Matt L further detailing variance with relation to the signal power: variance in the time domain versus variance in frequency domain
Update:
Dilip stated this all much more elegantly and concisely in the comments, paraphrased here: The area under the power spectral density function of a WSS process equals the value of the autocorrelation function at offset $\tau=0$. For a zero-mean WSS process, this value is the variance. Therefore deducing the power spectral density from knowledge of the variance is equivalent to asking for a curve given that we only know the area, which we know is not possible.