I have the following question :
$A,B,C$ are independent, $P(A)=0.2,P(B)=0.5,P(C)=0.8$, $N$ is a random variable that counts the number of incidents of $A,B,C$ that happened calculate Var(N)
I used indicators to calculate it yet, for some reason I get that $Var(N)<0$ which is wrong.
This is what I did:
$$Var(N)=E(N^2)-E(N)^2$$
$$E(N)=E(I_A)+E(I_B)+E(I_C)=0.2+0.5+0.8=1.5 \implies E(N)^2=2.25$$
$$E(N^2)=E(I_A^2)+E(I_B^2)+E(I_C^2)+E(I_A*I_B)+E(I_A*I_C)+E(I_B*I_C)$$
We know that by $E(N)$ that : $E(I_A^2)+E(I_B^2)+E(I_C^2)=E(I_A)+E(I_B)+E(I_C)=0.2+0.5+0.8=1.5$
since independent $$E(I_A*I_B)+E(I_A*I_C)+E(I_B*I_C)=0.2*0.5+0.2*0.8+0.8*0.5=0.66$$
Therefore we get that : $E(N^2)=1.5+0.66=2.16$
$$Var(N)=E(N^2)-E(N)^2=2.16-2.25=-0.09???$$
Any ideas what I did wrong I can't figure it out.
Thanks in advance :)