I am trying to model the timing of a series of events where event $Y$ takes place right after the latest of $X_1, X_2, ..., X_n$, where $n \in \{2, 3, 4\}$
For this I am assuming $$ \begin{align} X_i &\sim \mathcal{N}(\mu_i, \sigma_i)\\ Y &= \max(X_1, ..., X_n). \end{align} $$
It is relatively easy to prove that the CDF is $$ F_Y(y) = \prod_i F_{X_i}(y). $$ From this, the PDF can be calculated via chain rule.
Since I intend to approximate $Y$ by a Gaussian distribution, I am interested in its mean and variance. Unfortunately, for the cases $n=3$ and $n=4$, I don't have any good ideas for how to tackle it.
For the case $n=2$, I used formula 10,011.3 from Owen and 8.259.2 from Gradshteyn and Ryzhik.
Using \begin{align} \delta &= \frac{\mu_1-\mu_2}{\sqrt{\sigma_1^2+\sigma_2^2}}, \\ \phi(x) &= \frac{1}{\sqrt{2\pi}} \exp(-\frac{x^2}{2}) \end{align}
the mean and the variance are \begin{align} \mathbb{E}[Y] &= \frac{\mu_1+\mu_2}{2} + \frac{\mu_1-\mu_2}{2}\mathrm{erf}\left(\frac{\delta}{\sqrt{2}}\right) + \phi(\delta)\sqrt{\sigma_1^2 + \sigma_2^2} \\ \mathrm{Var}(Y) &= \mathbb{E}[Y^2] - \mathbb{E}[Y]^2 \\ &= \frac{\sigma_1^2 + \sigma_2^2 + \mu_1^2 + \mu_2^2}{2} + \frac{\sigma_1^2 - \sigma_2^2 + \mu_1^2 - \mu_2^2}{2} \mathrm{erf}\left(\frac{\delta}{\sqrt{2}}\right) \\ &\phantom{={}} + \phi(\delta)(\mu_1 + \mu_2)\sqrt{\sigma_1^2 + \sigma_2^2} - \mathbb{E}[Y]^2 \end{align}
For this simpler case of $n=2$, I tried throwing the integrals at Mathematica, but without luck (I have never used it before, so it might also be on me), therefore I didn't even try it with larger $n$.
This is very closely related to this question, but I am interested in the moments of the distribution for $n=3$ and $n=4$.
Since this is in a machine-learning context, I'm also considering approximations. One such idea is to calculate $\max(X_1, X_2, X_3) = \max(\max(X_1, X_2), X_3)$ and approximate $\max(X_1, X_2)$ with a Gaussian using the calculations for $n=2$, but this will make the result depend on the order of the calculations.
Any help in getting closed form solutions or good approximations is greatly appreciated!