Let $f_2(t)$ and $f_3(t)$ be density functions associated to distributions $F_2(t)$ and $F_3(t)$. I have the following formula, $$\frac{(f_2(t)+f_3(t))/2}{(1-(F_2(t)+F_3(t))/2)}=\lambda$$ where this equations holds $\forall t$ and $\lambda$ is some constant.
Before I begin to analyze this, I was wondering if it is well-known what class of distributions $F_2$ and $F_3$ satisfy this equation.
Two observations: Clearly, $F_2=F_3$ exponential with parameter $\lambda$ works, while $F_2,F_3$ exponential with parameters $\lambda_1 \ne \lambda_2$ does not work.
