I toss a fair coin $N$ times and $X$ is a RV that denotes number of transitions from heads to tails or from tails to heads in consecutive tosses. I have to find the expected value of $X$.
For this I have to find $P(X = k)$ where $k$ can vary from $1$ to $N-1$. I don't know how to handle this problem. I would be thankful for any help.
Usually such questions are solved using indicator random variables. Let $\mathbb{1}_{i}$ denote the indicator random variable for the transition from toss number $i$ to $i + 1$. Thus, $\mathbb{E}[\mathbb{1}_{i}] = P(\text{transition in } (i + 1)^{\text{th}} \text{ toss}) = \frac{1}{2}$ for all $i$.
Thus, we have $\displaystyle X = \sum_{i = 1}^{N-1} \mathbb{1}_i$. Hence, $\displaystyle \mathbb{E}[X] = \mathbb{E} \left[ \sum_{i = 1}^{N-1} \mathbb{1}_i \right] = \sum_{i = 1}^{N-1} \mathbb{E}[\mathbb{1}_i] = \sum_{i = 1}^{N-1} \frac{1}{2} = \frac{N - 1}{2}$.