The exponential distribution has a mode of $0$, which, according to Wikipedia, means that $0$ "is the value that is most likely to be sampled". This is not what I would expect, given that the exponential distribution "describes the time between events in a Poisson process".
So, say me getting phone calls is a Poisson process and I get a call every hour on average. How is it that $0$ is the most likely sample? My intuition tells me that $0$ is one of the most unlikely samples, or at least much less likely than some number around $1$ hour.
Best Answer
An exponential distribution is that of a continuous random variable. All particular values it can take have probability mass of zero.
The mode of a continuous random variable is not the point where its probability is most massive. It is where it is most dense.
The probability density function of an exponential random variable, $X$ with rate $\lambda$, is:
$$f_X(x)=\lambda \mathsf e^{-\lambda x} \mathbf 1_{x\in[0;\infty)}$$
This is at its maximum when $x=0$. Hence the mode of the distribution is: $0$.