I have another question regarding the Indicator Function, namely understanding the following equality:
$ 1 - E[\prod \limits_{i=1}^n (1 - 1_{A_i})] = \sum_{k=1}^{n} (-1)^{k+1} \sum_{1 \leq i_1 < ... < i_k \leq n} E[1_{A_{i_{1}}} * ... * 1_{A_{i_{k}}}] $
I am a bit unused to the summation sign on the right side. Given we would have to consider two events $A_1$ and $A_2$ only, is it correct that for the right hand side this would amount to:
$E[1_{A_1}]-E[1_{A_1}*1_{A_2}]$ ?
Thanks
No. Given only two events, the right-hand side would reduce to $$ E[1_{A_1}]+E[1_{A_2}]-E[1_{A_1}1_{A_2}]. $$ Given three events, it becomes $$ E[1_{A_1}]+E[1_{A_2}]+E[1_{A_3}]-E[1_{A_1}1_{A_2}]-E[1_{A_1}1_{A_3}]-E[1_{A_2}1_{A_3}]+E[1_{A_1}1_{A_2}1_{A_3}]. $$
Given an arbitrary number of events, the right-hand side will have one term for every nonempty subset of the set of events. Each of these terms will consist of the expectation of the product of the indicator functions of the events in the subset, taken with a positive sign if the number of events in the subset is odd, negative if it is even. The way this is expressed in the summation notation in your question is that the outer sum, over $k$, selects the number of events in the subset, while the inner sum chooses $k$ events out of the $n$ given.