TABLE II.

PHA formulas.

No pileup
λ = ∫S(E)dE	$\left\{0<E_1<\dots <E_{\mathrm{nbins}}\right\}$	With pileup
Photon number spectrum	S(E)	Srec(E) = λprec(E)
Normalized spectrum	p(E) = S(E)/λ	$p_{\mathrm{rec}}\left(E\right)=\sum _{k=0}^{\infty }\frac{{\eta }^k}{k!}{e}^{-\eta }\left(p^{\left(k\right)}\ast p\right)$
Bin probabilities	$P_k={\int }_{E_{k-1}}^{E_k}p\left(E\right)\mathrm{d}E$	$P_{\mathrm{rec},k}={\int }_{E_{k-1}}^{E_k}{p}_{\mathrm{rec}}\left(E\right)\mathrm{d}E$
Expected total counts	$〈N〉=\lambda$	$〈N_{\mathrm{rec}}〉=\frac{\lambda }{1+\eta }$
Variance total counts	var(N) = λ	$\mathrm{var}\left(N_{\mathrm{rec}}\right)=\frac{\lambda }{{\left(1+\eta \right)}^3}$
Nonpoisson factor	D = 0	$D=\mathrm{var}\left(N_{\mathrm{rec}}\right)-〈N_{\mathrm{rec}}〉$
Expected bin counts	$〈N_k〉=\lambda P_k$	$〈N_{\mathrm{rec},k}〉=〈N_{\mathrm{rec}}〉P_{\mathrm{rec},k}$
Variance bin counts	var(Nk) = λPk	$\mathrm{var}\left(N_{\mathrm{rec},k}\right)=〈N_{\mathrm{rec}}〉P_{\mathrm{rec},k}+D{P}_{\mathrm{rec},k}^2$
Covariance bin counts	0	cov(Nj,Nk)j≠k = Prec,jPrec,kD