We instantiate the general ROLL framework (\Cref{sec:roll-formulation}) with
Gaussian-distributed scores.  Each class's scores are modelled as
$\mathcal{N}(\mu_k, \sigma_k^2)$, fitted by maximum-likelihood estimation.

\paragraph{Forward pass.}

The MLE parameters are:
\begin{equation}
  \mu_k = \frac{1}{|\mathcal{B}_k|}\sum_{\mathbf{x}_j \in \mathcal{B}_k} f_\theta(\mathbf{x}_j),
  \qquad
  \sigma_k^2 = \frac{1}{|\mathcal{B}_k|}\sum_{\mathbf{x}_j \in \mathcal{B}_k}(f_\theta(\mathbf{x}_j)-\mu_k)^2
\end{equation}

The CDF and quantile function of $\mathcal{N}(\mu,\sigma^2)$ are:
\[
  F(x) = \tfrac{1}{2}\!\left[1 + \operatorname{erf}\!\left(\tfrac{x-\mu}{\sigma\sqrt{2}}\right)\right],
  \qquad
  F^{-1}(p) = \mu + \sigma\sqrt{2}\,\operatorname{ierf}(2p-1)
\]
where $\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,\mathrm{d}t$ is the Gauss
error function and $\operatorname{ierf}$ its functional inverse.
Substituting into the general ROLL objective \eqref{eq:roll-tpr-at-fpr}:
\begin{align}
  \tau &= \mu_0 + \sigma_0\sqrt{2}\,\operatorname{ierf}(1-2\alpha) \nonumber \\
  \mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{GAUSSIAN}}(f_\theta(\mathcal{B});\,\alpha)
    &= \tfrac{1}{2}\!\left[1 + \operatorname{erf}\!\left(\tfrac{\tau-\mu_1}{\sigma_1\sqrt{2}}\right)\right]
  \label{eq:roll-tpr-at-fpr-gaussian}
\end{align}

\paragraph{Gradient derivation.}

We derive the gradient of \eqref{eq:roll-tpr-at-fpr-gaussian} with respect to
$f_\theta(\mathbf{x}_i)$, instantiating the general structure~\eqref{eq:roll-gradient}
for the Gaussian case. Throughout, $\tau = \hat{F}_0^{-1}(1-\alpha) =
\mu_0 + \sigma_0\sqrt{2}\,\operatorname{ierf}(1-2\alpha)$.

\paragraph{Case $y_i = 1$.}

The score $f_\theta(\mathbf{x}_i)$ enters through $\mu_1$ and $\sigma_1$.
Differentiating $\hat{F}_1(\tau) = \frac{1}{2}\bigl[1+\operatorname{erf}\!\bigl(\frac{\tau-\mu_1}{\sigma_1\sqrt{2}}\bigr)\bigr]$:

\[
  \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial \mu_1} =
  -\frac{1}{\sigma_1\sqrt{2\pi}}\exp\!\left(-\frac{(\tau - \mu_1)^2}{2\sigma_1^2}\right)
\]

\[
  \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial \sigma_1} =
  \frac{\mu_1 - \tau}{\sigma_1^2\sqrt{2\pi}}\exp\!\left(-\frac{(\tau - \mu_1)^2}{2\sigma_1^2}\right)
\]

Combined with $\frac{\partial \mu_1}{\partial f_\theta(\mathbf{x}_i)} = \frac{1}{|\mathcal{B}_1|}$
and $\frac{\partial \sigma_1}{\partial f_\theta(\mathbf{x}_i)} = \frac{f_\theta(\mathbf{x}_i)-\mu_1}{|\mathcal{B}_1|\sigma_1}$,
the chain rule gives:

\begin{equation}
  \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial f_{\theta}(\mathbf{x}_i)}
  = -\frac{1}{|\mathcal{B}_1|\,\sigma_1\sqrt{2\pi}}
    \exp\!\left(-\frac{(\tau - \mu_1)^2}{2\sigma_1^2}\right)
    \left(1 + \frac{(\tau - \mu_1)(f_\theta(\mathbf{x}_i) - \mu_1)}{\sigma_1^2}\right)
  \label{eq:gauss-grad-y1}
\end{equation}

\paragraph{Case $y_i = 0$.}

The score $f_\theta(\mathbf{x}_i)$ affects the loss only through $\tau$. By the chain rule:

\[
  \frac{\partial \hat{F}_1(\tau)}{\partial f_\theta(\mathbf{x}_i)}
  = \underbrace{\frac{\partial \hat{F}_1(\tau)}{\partial \tau}}_{\text{PDF of class 1 at }\tau}
    \cdot \frac{\partial \tau}{\partial f_\theta(\mathbf{x}_i)}
\]

The first factor is the Gaussian PDF at $\tau$:
\[
  \frac{\partial \hat{F}_1(\tau)}{\partial \tau}
  = \frac{1}{\sigma_1\sqrt{2\pi}}\exp\!\left(-\frac{(\tau - \mu_1)^2}{2\sigma_1^2}\right)
\]

For the second factor, since $\tau = \mu_0 + \sigma_0\sqrt{2}\,\operatorname{ierf}(1-2\alpha)$:
$\frac{\partial\tau}{\partial\mu_0}=1$ and
$\frac{\partial\tau}{\partial\sigma_0}=\frac{\tau-\mu_0}{\sigma_0}$.
Combined with $\frac{\partial\mu_0}{\partial f_\theta(\mathbf{x}_i)}=\frac{1}{|\mathcal{B}_0|}$
and $\frac{\partial\sigma_0}{\partial f_\theta(\mathbf{x}_i)}=\frac{f_\theta(\mathbf{x}_i)-\mu_0}{|\mathcal{B}_0|\sigma_0}$:
\[
  \frac{\partial\tau}{\partial f_\theta(\mathbf{x}_i)}
  = \frac{1}{|\mathcal{B}_0|}\!\left(1+\frac{(\tau-\mu_0)(f_\theta(\mathbf{x}_i)-\mu_0)}{\sigma_0^2}\right)
\]

Therefore:
\begin{equation}
  \dfrac{\partial \hat{F}_1(\hat{F}_0^{-1}(1-\alpha))}{\partial f_\theta(\mathbf{x}_i)}
  = \frac{1}{|\mathcal{B}_0|\,\sigma_1\sqrt{2\pi}}
    \exp\!\left(-\frac{(\tau - \mu_1)^2}{2\sigma_1^2}\right)
    \left(1 + \frac{(\tau - \mu_0)(f_\theta(\mathbf{x}_i) - \mu_0)}{\sigma_0^2}\right)
  \label{eq:gauss-grad-y0}
\end{equation}

Substituting \eqref{eq:gauss-grad-y1} and \eqref{eq:gauss-grad-y0} into \eqref{eq:roll-gradient}:
\begin{equation}
  \frac{\partial \mathcal{L}_{\text{ROLL-TPR@FPR}}^{\text{GAUSSIAN}}}{\partial f_\theta(\mathbf{x}_i)} =
  \begin{cases}
    \dfrac{-1}{|\mathcal{B}_1|\,\sigma_1\sqrt{2\pi}}
    \exp\!\left(-\dfrac{(\tau-\mu_1)^2}{2\sigma_1^2}\right)
    \!\left(1 + \dfrac{(\tau-\mu_1)(f_\theta(\mathbf{x}_i)-\mu_1)}{\sigma_1^2}\right)
      & \text{if } y_i = 1 \\[18pt]
    \dfrac{1}{|\mathcal{B}_0|\,\sigma_1\sqrt{2\pi}}
    \exp\!\left(-\dfrac{(\tau-\mu_1)^2}{2\sigma_1^2}\right)
    \!\left(1 + \dfrac{(\tau-\mu_0)(f_\theta(\mathbf{x}_i)-\mu_0)}{\sigma_0^2}\right)
      & \text{if } y_i = 0
  \end{cases}
  \label{eq:gauss-grad-combined}
\end{equation}