I am observing two continuous time-series where at every instant in time I may observe a unary event. That is, for each sequence, say $S_1$, I have a data set comprised of $S_1 = (t_0, t_1, ..., t_m)$ where $t_i \in (0, T) \subset \mathbb{R}$. So if I observed the event at some time $t_i$ along my data stream $1$ then I add $t_i$ to $S_1$

I'de like to say that events happen in one sequence more than another with level of confidence.

Can I just compute the mean as follows?

$$ \mu_1 = \int_0^T \delta(t \in S_1) dt = \frac{|S_1|}{T} $$

Where $|S_1|$ gives the cardinality of $S_1$. Also, does it make sense to compute variance as: $$ \sigma_1^2 = \int_0^T (\delta(t \in S_1)-\mu_1)^2 dt = \frac{|S_1|}{T} - \left(\frac{|S_1|}{T}\right)^2 $$

So, if I want to create error bars I would guess I would have: $$ \mu \pm \frac{2\sigma_1}{\sqrt{T}} $$

I'm assuming this is not correct because the value changes depending on my units of time. If I use seconds, nanoseconds, or picoseconds I get a different mean and variance. I would hope that as I watch these sequences for a longer and longer period of time my uncertainty about their emission probabilities would converge to zero.


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