4.7 Sampling & Confidence

Not So Strong


For each positive integer \(i\), let the random variable \(X_i\) be defined as:

\(\begin{align}\Pr[X_i= \pm i\cdot 2^i] = f(i)\\\Pr[X_i=0]= 1-2f(i)\end{align}\)

where \(0\leq f(i) \leq \frac{1}{2}\).

Let \(S_n:=X_1+X_2+\ldots + X_n\).

What is \(E[S_n]\)?

Exercise 1

\(E[X_i]=-i2^i\cdot f(i) + 0\cdot(1-2f(i)) + i2^i\cdot f(i) = 0\). By linearity of expectation \(E[S_n]= 0 \)