I have been having some trouble with "Indicator random variables". Here's an example:
It is known that $P \left( A \right) =0.7 $, $P \left( B\right) =0.6 $, and $P \left( A \cup B \right) =0.8 $ Find the Covariance between the Indicator Random Variables $1_{A}$ and $1_{A \cap B}$.
$P \left( A \cup B \right) = P\left( A \right) + P\left( B \right) – P \left( A \cap B \right) \Rightarrow P \left( A \cap B \right)=0.5 $
For the rest… Any ideas would be appreciated.
Best Answer
Let $\Omega$ denote the set of possible outcomes. (Note that $A,B \subseteq \Omega$.)
We have $$ \mathbb{E}[1_A] = \int_\Omega 1_{\omega \in A} d\mathbb{P}(\omega) = \mathbb{P}[A]. $$ Note that $1_A^2 = 1_A$ therefore $\mathbb{E}[1_A^2] = \mathbb{E}[1_A] = \mathbb{P}[A]$.
Find the variance, do the same for $\mathbb{E}[1_{A\cap B}]$ and plug into the definition of covariance.