Lesson 3

Problem. Consider an experiment with success probability \(\frac{1}{3}\). Repeat the experiment \(n\) times, and let the random variable \(\alpha_{i}\) be the indicator of success for the \(i\)-th repetition (that is, \(\Pr[\alpha_{i}=1]=\frac{1}{3}\) and \(\Pr[\alpha_{i}=0]=\frac{2}{3}\)). Let also \(\beta\) be the random variable equal to the proportion of successes among \(n\) repetitions: \(\beta=\frac{1}{n}(\alpha_{1}+\dotsb+\alpha_{n})\). Compute \(\operatorname{E}[\beta]\).