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Here's an interesting/natural observation which might be useful. Let $U$ be a uniform$(0,1/2)$ random variable, so that $U$ has density function $f(x)=2$, $0<x<1/2$, and define a function $Y$ by $Y(x)=(1/2-x)Q(x)$, $0<x<1/2$. Then, $M=1/12 + \frac{1}{2}{\rm E}(Y^2 ) - {\rm E}^2 (Y)$. (Interestingly, a uniform$(0,1)$ random variable has variance equal to $1/12$.)