Let $X,Y$ two random variable of density $f_X$ and $f_Y$. I was wondering under which condition (independence,...) the following statement is true. For $\lambda \in ]0,1[$ is $\lambda X + (1-\lambda) Y$ of density $\lambda f_X + (1-\lambda) f_Y$ ? Normality is a first answer ! But is it true in other contexts ?
I can't find any specifics about this question, since it's not exactly a mixture distribution. The following link Prove density function of $Z=\lambda X+(1-\lambda)Y$ is $\lambda f_X(z)+(1-\lambda)f_Y(z)$ where $\lambda\in(0,1)$ is pointing the confusion, but no clear answer.
Thank your for your help