Not So Strong
For each positive integer \(i\), let the random variable \(X_i\) be defined as:
where \(0\leq f(i) \leq \frac{1}{2}\).
Let \(S_n:=X_1+X_2+\ldots + X_n\).
What is \(E[S_n]\)?
\(E[X_i]=-i2^i\cdot f(i) + 0\cdot(1-2f(i)) + i2^i\cdot f(i) = 0\). By linearity of expectation \(E[S_n]= 0 \)