So the problem states that let n>=2 be an integer, Consider a bitstring $b_1$; $b_2$;...;$b_n$ 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<=i< n for which $b_i$$b_{i+1}$ = 0.
What is the expected value E(X) of the random variable X? [Use indicator random variables] The answer for this is ${3(n-1)\over4}$
My first shot at this!
I am particularly confused on how to properly use the indicator functions/variables. For this question, I'm assuming it should be P(X=1) = 1/2 for the indicator variable.
There's also this constraint of bi*bi+1 = 0, so I'm assuming one of bi or bi+1 has to be 0. I'm not sure exactly how to properly assess the data given in the problem and apply the Expected value formula. It would be greatly appreciated if somebody could help me break down this problem so I have a better understanding :)
Let $$X_i = \begin{cases} 1 &, \text{ if } b_ib_{i+1}=0 \\0 &, \text{ if }b_ib_{i+1} =1 & \end{cases}$$
where $i \in \{1, \ldots, n-1\}$.
We have $E[X_i] = \frac34$ since I just need one of the bit to be $0$.
Hence $E[X] =E[\sum_{i=1}^{n-1} X_i]= \frac34(n-1)$.