a distribution function If there are two multivariate independent gaussian variables, with their distribution function  $F_1$ and $F_2$ then by what conditions the function $F(x):=F_1(x)+F_2(x)-F_1(x)F_2(x)$ is a distribution function?
 A: Consider the problem without the word gaussian included in it, and even more
to the point, without the word multivariate included in it. If $A$ and $B$
are (ordinary, single or univariate) independent 
random variables with distribution functions $F_1(\cdot)$ and $F_2(\cdot)$ respectively and $C$ is defined to be $\min\{A,B\}$, 
then
$$\begin{align}
F_C(x) &= P\{C \leq x\}\\
&= P\left(\{A \leq x\} \cup \{B \leq x\}\right)\\
&= P\{A \leq x\} + P\{B \leq x\} - P\left(\{A\leq x\}\cap \{B\leq x\}\right)
& \text{standard formula}\\
&= P\{A \leq x\} + P\{B \leq x\} - P\{A\leq x\}P\{B\leq x\} &\text{by independence}\\
&= F_1(x) + F_2(x) - F_1(x)F_2(x).
\end{align}$$
So, for univariate independent random variables with distribution
functions $F_1(\cdot)$ and $F_2(\cdot)$, the distribution of
the minimum of the two random  variables has the desired form
$$F_{\min}(x) =F_1(x) + F_2(x) - F_1(x)F_2(x)$$
and gaussianity is not needed for the result to hold.
For multivariate random variables (also called random
vectors), you should give some
details as to what is meant by $F_1(x)$ and $F_2(x)$, and
what is meant by the $F_1(x)+ F_2(x)$ when the two random
vectors are of different lengths.
A: For any function to be a valid distribution function, it needs to hold 4 properties,namely 


*

*distribution function F is (not necessarily strictly) monotone non-decreasing.

*F is right-continuous.

*$\lim\limits_{x \to-\infty}F = 0$ 

*$\lim\limits_{x \to\infty}F = 1$ 


Now we rewrite the given expression as $F(x):=1-(1-F_1(x))(1-F_2(x))$ and we can easily observe the 4 properties in $F(x)$ given that $F_1(x)$ and $F_2(x)$ are valid distribution function.
A: Thanks. The phenomena being modelized corresponds to the sampling from two multivariate gaussian distributions (with different mean vectors, but the same spherical variance matrix). So, given a point in this real n-dimensional space, what is the probability of this point belonging to the first distribution or the second one, taking into account that it may also belong to both of them. At what point would it be reasonable to consider the independency of the two gaussian variables? 
