A question about the probability that the max of N dependent gaussian random variables with a deterministic covariance matrix exceed a constant gamma The problem is given as follows,
I have a set of gaussian random variables $\{T_1,T_2,...T_N\}$, the mean value for each random variable is $\mu_n=k_n F$, where $k_n$ is an unknown but deterministic integer and $k_n \in [0,K]$.
$F$ is a function that I want to optimize and $F>0$.
The covarianve for each of them is $1$.
The covariance matrix $M$ of the $N$ gaussian random variables is unknown but deterministic.
There is a probability that is defined as
\begin{equation}
P=Pr\{\max\{T_1,T_2,...T_N\}>\gamma\}
\end{equation}
The question is "Can I say that $P$ can obtain its maximum when $F$ achieves its maximum?"
 A: The random vector variable $\mathbf{T} =(T_1,...,T_N)$ follows the multivariate normal distribution
$\mathcal{N}_N \left(\mathbf{k}F;\mathbf{M}  \right)$ where $\mathbf{k}$ is the vector $(k1,...,k_N)$.
We have: $\mathbf{T} = \mathbf{k}F+\mathbf{Z}$ with $\mathbf{Z}$ follows the distribution $\mathcal{N}_N \left(\mathbf{0}_N;\mathbf{M}  \right)$ ($\mathbf{0}_N$ is the vector $0$ of dimension $N$). Let's denote $f_{\mathbf{Z}}(\mathbf{z})$ the PDF of $\mathbf{Z}$.
\begin{align}
P(F) &= 1- P(\max\{T_1,...,T_N\} \le \gamma) \\
  &= 1- P(T_1\le \gamma,...,T_N\le \gamma) \\
  &= 1- P(Z_1\le \gamma - k_1F,...,Z_N\le \gamma-k_NF) \\
  &= 1- \int_{\{\mathbf{z}\in \mathcal{D}(F)\}} f_{\mathbf{Z}}(\mathbf{z})d\mathbf{z} \tag{1}\\
\end{align}
The integral is  calculated over the zone $\{\mathbf{z}\in \mathcal{D}(F)\}$ where $\mathcal{D}(F)$ is defined as
$$\mathcal{D}(F)= (-\infty,\gamma-k_1F ) \times \ldots \times (-\infty,\gamma-k_NF )$$
For 2 values $F_1$ and $F_2$ such that $F_1 > F_2$, we have $\gamma-k_nF_1 < \gamma-k_nF_2 $ for all $n =1,...,N$. So
$$(-\infty,\gamma-k_nF_1 ) \subset (-\infty,\gamma-k_nF_2 )$$
or
$$\mathcal{D}(F_1) \subset\mathcal{D}(F_2)  \tag{2}$$
From $(2)$, we deduce $\int_{\{\mathbf{z}\in \mathcal{D}(F_1)\}} f_{\mathbf{Z}}(\mathbf{z})d\mathbf{z} < \int_{\{\mathbf{z}\in \mathcal{D}(F_2)\}} f_{\mathbf{Z}}(\mathbf{z})d\mathbf{z}$.
Hence, we can deduce from $(1)$ that
$$ P(F_1) > P(F_2) $$
Conclusion: if $k_n \ge 0$, $P$ obtain its maximum when $F$ achieves its maximum.
