Question: 2017 Fall Final - 19
Let $n \geq 2$ be an integer. Consider a bitstring $b_1,b_2,\dots,b_n$ of length $n$, in which each
bit $b_i$ is 0 with probability 1/2, and 1 with probability 1/2 (independent of all other bits).
Define the random variable $X$ to be the number of indices $i$ with $1 \leq i < n$ for which $b_i \neq b_{i+1}$.
What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
Hint: Use indicator random variables.
Define the random variable $X$ to be the number of indices $i$ with $1 \leq i < n$ for which $b_i \neq b_{i+1}$.
What is the expected value $\mathbb{E}(X)$ of the random variable $X$?
Hint: Use indicator random variables.
(a)
$\frac{n}{2}$
(b)
None of the above.
(c)
$\frac{n-1}{2}$
(d)
$\frac{n+1}{2}$