Averaging inverse CDFs Suppose I have two distributions $P$ and $Q$ on the line that admit  well defined inverse cumulative distribution functions $F^{-1}_P$ and $F^{-1}_Q$. 
I define an "average" distribution $A$ as the distribution whose inverse CDF is given by 
$$ F^{-1}_A = (1/2)(F^{-1}_P + F^{-1}_Q) $$
Is this distribution $A$ well known in any sense ? Can this operation be expressed directly in terms of $P$ and $Q$ without needing to go through the inverse CDFs ? 

(This is not a full answer.)
Conjecture: The CDF $F_A$ is the CDF of the random variable $\frac12(X+Y)$, where the random variable $X$ has distribution $P$, the random variable $Y$ has distribution $Q$ and the random couple $(X,Y)$ is maximally coupled (for example, if $P$ and $Q$ are square integrable then $E(XY)$ is maximal).
Easy fact supporting the conjecture: The CDF $F_A$ is the CDF of the random variable $\frac12(F_P^{-1}(U)+F_Q^{-1}(U))$, where the random variable $U$ is uniform on $(0,1)$. Note that the random variables $F_P^{-1}(U)$ and $F_Q^{-1}(U)$ have distribution $P$ and $Q$ respectively.
