hello I have two independent variable P and Q. They are both non-negative. Now I define two new variables on them:
The first variable $$R_1=\alpha P+(1-\alpha)Q.$$ Since P and Q are independent, so $$Var(R_1)=\alpha^2 Var(P)+(1-\alpha)^2Var(Q)$$
The second variable R2 is a sort of compound variable: there is a probability of $\alpha$ that we get P and $1-\alpha$ probability to get Q. I work out the variance as $$Var(R_2)=\alpha E(P^2)+(1-\alpha)E(Q^2)-(\alpha E(P)+(1-\alpha)E(Q))^2$$
My intuition is that $$Var(R_2) \geq Var(R_1).$$ I wonder if anyone could help with the proof of the intuition above?
Here is an example. Let $P=Q=(10,0.5;0,0.5)$ which means there is 0.5 probability to get 10 and 0.5 probability to get 0. Let $\alpha=0.5$. Then $R_1=(10,0.25;5,0.5;0,0.25)$ and $R_2=(10,0.5;0,0.5)$. We get $Var(R_1)=12.5$ and $Var(R_2)=25$.
I tried a couple of other examples and they all show that $Var(R_2)>Var(R_1)$.