A message is sent over a communication channel. The message is made up of $n$ symbols, each of which may be $0$ and $1$. Each is equiprobable and independent of the other. Find the expectation and variance of the random variable $X$ which corresponds to the number of changes in the symbols (for example, if $0101$ is the message, the number of changes is $3$ in the message.
2026-03-25 04:38:55.1774413535
Number of changes in adjacent symbol in a binary message
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Hint: Find expectation of an indicator random variable $X_i$, which takes the value $1$ if change in symbols occurs between $i^{th}$ to $(i+1)^{th}$, and $0$ otherwise. This is just the probability of a change of symbol from $i^{th}$ to $(i+1)^{th}$ index, which is easy to calculate.
Then use $X = \sum X_i$ and linearity of expectation to get $E(X)$.