Runs in coin tosses

You have a biased coin i.e., each coin toss is a head with probability \(p\) and is a tail with probability \(1-p\). Let \(X\) be the number of runs in \(n\) independent tosses of this coin. Here, runs are consecutive tosses with the same side (heads or tails).

  • Compute \(E[X]\), the expectation of \(X\).

  • Show that \(Var[X] \le 4 n \cdot p(1-p)\), where \(Var[X]\) is the variance of \(X\).

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