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→Results
\left( \begin{matrix} n\\k \end{matrix} \right)
\left( \begin{matrix} n\\k \end{matrix} \right)
(1-\alpha-\beta)^{m-n} \alpha^{n-k} \beta^{k}
(1-\alpha-\beta)^{m-n} \alpha^{n-k} \beta^{k}
\ G\left( (m-n) + (n-k)p + kr,\ (m-n)\sigma_1^2 + (n-k)\sigma_2^2 + k\sigma_3^2 \right)
\ G\left( (m-n) + (n-k)p + kr,\ \sigma_0^2 + (m-n)\sigma_1^2 + (n-k)\sigma_2^2 + k\sigma_3^2 \right)
</math>
</math>
[[Image:HPD_OnePEdist.png|thumb|300px|Distributions of primary and secondary photoelectron signals (noise included) for one incident primary photoelectron.]]
[[Image:HPD_OnePEdist.png|thumb|300px|Distributions of primary and secondary photoelectron signals (noise included) for one incident primary photoelectron.]]
The corrected distribution of secondaries, made up of two Gaussians to account for its slight asymmetry, allows the model to fit the measured spectra. First, the following parameters were fixed from HPD/amplifier specifications and no-light distribution ("pedestal") measurement:
* Total gain, including amplifier and oscilloscope waveform mean rescaling: 1.9264 × 10<sup>5</sup> V/p.e.;
* Pedestal position: -3.6606 × 10<sup>-3</sup> V;
The parameters relating to the shape of the primary and secondary electron distributions, as well as an additional Gaussian noise factor (''σ''<sub>0</sub>) were allowed to vary. A fit with these degrees of freedom was performed on three distributions from drastically different light levels (corresponding to roughly 64, 530 and 973 primary detected photons.) The following model parameters produced reasonably good fits (''χ''<sup>2</sup><2.0) for all three distributions.
{| border="0" cellpadding="20"
|
{| style="text-align:center" border="1" cellspacing="0" cellpadding="5"
|- style="background:#F1F2FF"
! colspan="2"| Gaussian 1
! colspan="2"| Gaussian 2
! Units
|-
| ''p'' || 107.0 || ''r'' || 179.5 || p.e.
|-
| ''α'' || .000899 || ''β'' ||.000299 || -
|-
| ''σ''<sub>2</sub> || 35.83 || ''σ''<sub>3</sub> || 119.2 || p.e.
|-
|}
||
Widths of primary peaks and overall noise:
* ''σ''<sub>0</sub> = 13.7 p.e.
* ''σ''<sub>1</sub> = 0.881 p.e.
|}
[[Image:HPD_3dist.png|frame|Distributions on a logarithmic plot from runs with different light levels (roughly 64, 530 and 973 primary detected photons.) Units are arbitrary and histograms have been rescaled for comparison.]]
Fits with far better agreement were found when the above-listed parameters were free to vary individually for each distribution. Similar and more drastic problems occurred when the simpler one-Gaussian model was used for the distribution of secondary photo-electrons. Evidently, the inaccuracy in its shape is compounded with more convolutions necessary in the high-photon count spectra. Thus, we suspect that the two-Gaussian approximation of this asymmetric distribution is yet not full adequate. However, this already computationally-intensive model (note the three summations) will be even more difficult with more Gaussian and intractable for an arbitrary shape since the convolutions will no longer be straightforward. Disagreements between the fit and data in the right-hand tails of the spectra may also be due to the truncation of the summation in ''n'' corresponding to the number of peaks present due to the convolution with primary and secondary distributions.