Suppose that the physical system is in generic state $|\psi\rangle$. Show that $\sum_{\lambda}p^2_{\lambda} = 1$ to an observable $O$, if and only if $\Delta O = 0$. ($\Delta O$ is a standard deviation).
Using the probability measure definition $p_{\lambda} = \langle\psi|P_{\lambda}|\psi\rangle$. Using hypothesis, $$\sum_{\lambda}\langle\psi|P_{\lambda}|\psi\rangle^2 = 1,$$ the fact that $\sum_{\lambda}\langle\psi|P_{\lambda}|\psi\rangle = 1$ and the definition $\Delta O = \sqrt{\langle O^2\rangle - \langle O\rangle^2 }$.
I tried $$\langle O^2\rangle - \langle O\rangle^2 = \langle\psi|O^2|\psi\rangle - (\sum_{\lambda}\lambda\langle\psi|P_{\lambda}|\psi\rangle)^2,$$ but dont know what else to do