## 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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