Show that every random variable $X$ can be written as $$X=\lambda Z_1+(1-\lambda)Z_2$$ for a discrete random variable $Z_1$, a continuous random variable $Z_2$, and a real value $\lambda$.
This exercise was proposed in class. I really don't know if the result is true.
I know the trivial cases ($X$ discrete and $X$ continuous), but need help in the general case. Can anybody help me?