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I am interested in the integral $P(f_i \leq f \leq f_j)$ where each function follows a normal distribution with distinct parameters.

With a bit of algebra, I managed to show that this integral is equivalent to $N_3([\mu, \mu_i, \mu_j], diag\{ \sigma^2, \sigma_i^2, \sigma_j^2\})$ where $N_3$ denotes the multinormal distribution of $3$ variables.

Written explicitly, we have $$P(f_i \leq f \leq f_j) = \frac{1}{4\sqrt{2\pi \sigma_i}}\int_{\mathbb{R}} \int_{z}^\infty \int_{-\infty}^z \exp\left\{ -\frac{1}{2}\left[\left(\frac{z-\mu}{\sigma}\right)^2 + \left(\frac{y-\mu_j}{\sigma_j}\right)^2 + \left(\frac{x-\mu_i}{\sigma_i}\right)^2\right]\right\}dxdydz$$

where $\mu, \mu_i, \mu_j, \sigma, \sigma_i, \sigma_j$ are constants. Wolfram is able to solve the first two iterates $dx$ and $dy$ to obtain

$$\begin{align*} P(f_i \leq f \leq f_j)&=\frac{1}{4\sqrt{2\pi \sigma_i}}\int_{-\infty}^\infty \exp\left(-\frac{(\mu-z)^2}{2\sigma}\right)\left(1+\text{Erf}\left[\frac{\mu_j-z}{\sqrt{2}\sqrt{\sigma_j}}\right]\right)\left(1-\text{Erf}\left[\frac{\mu_i-z}{\sqrt{2}\sqrt{\sigma_i}}\right]\right)dz\\[0.5em] &= \frac{1}{4\sqrt{2\pi \sigma_i}}\int_{-\infty}^\infty \exp\left(-\frac{(\mu-z)^2}{2\sigma}\right)\left(1+\text{Erf}\left[\frac{\mu_j-z}{\sqrt{2}\sqrt{\sigma_j}}\right]\right)\text{Erfc}\left[\frac{\mu_i-z}{\sqrt{2}\sqrt{\sigma_i}}\right]dz \end{align*} $$

But is unable to find a closed form for the last one.

I am wondering if there are other approaches I could be taking when solving this integral, for example making use of some nice properties of $\exp$.

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