Normalizing white noise to match pink noise's amplitude at a given frequency?
Okay, when discussing white noise or pink noise (or red noise or brown noise or flicker) or some other random process, there is this property called the power spectrum, in which white noise has a constant value for all frequencies.  But we integrate the power spectrum over all frequencies (from $-\infty$ to $+\infty$) to get power.  Integrating a constant, non-zero function over an infinite frequency range gives you infinite power.

So in reality, white noise does not exist.  (It puts out more power than the Sun.)

But if you bandlimit white noise to a finite bandwidth, you have a finite power.

Sampled signals (in discrete time) must have a finite bandwidth in order to be sampled properly.

If you were to synthesize bandlimited white noise by use of a random number generator that produces good random numbers with a uniform distribution between the values of $0$ and $\Delta$, you would have a mean of $\frac{\Delta}{2}$ and a variance of $\frac{\Delta^2}{12}$.  If you subtract from each uniform random number the mean of $\frac{\Delta}{2}$, then you would have the same variance, but a mean of zero.

Now if you were to add $N$ of those independent zero-mean uniform random numbers together, the variance would increase to $\frac{N \Delta^2}{12} \triangleq \sigma^2 $ and the distribution of the random numbers would closely approach that of a normal distribution (sometimes called Gaussian).

Now variance of a zero mean (or DC free) random process is the power, which is the area (or integral) of the power spectrum.  So from $-\frac{f_\mathrm{s}}{2}$ to $+\frac{f_\mathrm{s}}{2}$, you would have a flat power spectrum with an amplitude of $\frac{\sigma^2}{f_\mathrm{s}}$ so that the area or integral comes out to be $\sigma^2$.  (The amplitude in the power spectrum is zero for all $|f|&gt;\frac{f_\mathrm{s}}{2}$.  This is what makes it bandlimited.)

So now you have "white" noise that is finite energy.  When you run that through a pinking filter (is that what you are doing to make pink noise?), then you have to multiply that constant amplitude of $\frac{\sigma^2}{f_\mathrm{s}}$ times the magnitude-squared of the pinking filter to get the frequency-dependent amplitude of the pink noise.  Then you have to integrate that from from $-\frac{f_\mathrm{s}}{2}$ to $+\frac{f_\mathrm{s}}{2}$ to get the power of the pink noise.  Whatever is the ratio of that power to the power of the bandlimited white noise going in, you compute the square-root of the reciprocal, and that is the scaling constant to multiply times your pink noise to make it have the same power as the bandlimited white noise going in.