Is rms value of a positive varying DC voltage same with its mean value?
The rms voltage is given by 

\$\sqrt{\frac{1}{t_1-t_0}\int_{t_0}^{t_1}v^2(t) \mathrm dt}\$

The mean voltage is given by 

\$\frac{1}{t_1-t_0}\int_{t_0}^{t_1}v(t) \mathrm dt\$

As you can see, these are not the same, except in special cases. A pure positive dc voltage is one such special case. 

However, if the dc component of voltage is much bigger than any ac components, the rms and mean will be very close to each other. This could apply to the case of "noise with a huge offset".