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The question and answer below is related to the renewal process. I'm curious how the "I" changed to "H(T)" as indicated by the yellow boxes. Thanks for spending the time to look over my newbie question!

Question

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Answer

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By definition, the distribution $H(t)$ of the random variable $X$ is simply $\mathbb{P}(X < t)$. So, the indicator random variable $I$ is $1$ with probability $p = \mathbb{P}(X <T) = H(T)$, and is $0$ with probability $1-p$, i.e. it is a Bernoulli random variable, and its expectation is simply $\mathbb{E}(I) = p = H(T)$.

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