# Instantaneous Event Probability in Poisson Process

In a homogeneous Poisson process with rate $\lambda$, what is the probability of observing an event in an "instant," that is, an infinitesimally small interval of length dt? I have read that the Poisson rate function $\lambda(t)$ can be defined as the "instantaneous probability of observing a spike at each point in time." (http://www.stat.columbia.edu/~liam/teaching/neurostat-spr11/uri-eden-point-process-notes.pdf) But for a homogeneous process with $\lambda(t) = \lambda$, how can this be when it is possible that $\lambda > 1$?

The instantaneous probability of observing a spike between $t$ and $t + dt$ is $\lambda(t)dt$ (mind the $dt$ term). This can be noticed directly from the definition. For example with the homogenous Poisson process: $$P [(N(t+ \tau) - N(t)) = k] = \frac{e^{-\lambda \tau} (\lambda \tau)^k}{k!}$$
looking at $k = 1$ and $\tau = dt$ gives $P[dN(t)] = \lambda dt$.
• Thank you for asking me to clarify (and thank you for your help!) I am trying to define $X(t)$ that takes the value $1$ if there is an event at $t$, and 0 else. My goal is to find the expectation of $X(t)$. Is this question well-posed? This goes a bit beyond my undergraduate training in statistics. Commented Jan 12, 2013 at 14:21
• If you consider a poisson process, it makes no sense to consider an event at exactly $t$ because the probability will be zero. Similarly, if you have a gaussian variable $G$, the probability that $G$ be exactly equal to some real $x$ is zero. It is indeed more interesting to consider the amount $P[G \in [x, x + dx]] = f(x)dx$ where we define $f$ as the density function of $G$. Commented Jan 12, 2013 at 14:30