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I don't understand this sentence from this paper (around equation $5$):

The function $H(\tau)$ is the hazard function.

$H(\tau) = \frac{P_{\text{gap}}(g = \tau)}{\sum_{t=\tau}^{\infty} P_{\text{gap}}(g)}$

In the special case is where $P_{\text{gap}}(g)$ is a discrete exponential (geometric) distribution with timescale $\lambda$, the process is memoryless and the hazard function is constant at $H(\tau) = 1/\lambda$.

I understand that the discrete exponential distribution is the geometric distribution, but the two distributions have different parameters:

$$ X \sim \text{Geom}(p), \quad Y \sim \text{Expon}(\lambda) $$

In the discrete case, I think of $p$ as the probability of success and the probability of that $X$ takes value $i$ is the probability that we fail $i-1$ times before finally succeeding. But in the continuous case, I view $\lambda$ as a rate: the rate of successes helps us compute how long we wait until the first success.

Assuming "timescale" = "rate", what does it mean that a geometric distribution has a particular rate parameter?

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You can actually deduce the relationship by noting that the hazard rate for a Geometric distribution is constant and equal to $p$, the probability of success. Consequently, in order for the hazard rate of the discrete exponential $(1/\lambda)$ to equal $p$, $\lambda$ must equal $1/p$.

Although you view $\lambda$ as a rate in the continuous (Exponential) case, the Exponential can actually be parameterized in terms either of the rate or of the inverse of the rate, also known as the "scale parameter". (See the Wikipedia article on the Exponential distribution under "Alternative parameterization" for an example.) In both the Exponential and Geometric cases, the expected value of the variate is equal to the scale parameter (assuming $0$ is in the range of the Geometric,) so the rate at which the "event" occurs does equal $1/$ the scale. Since by far the most common use of hazard rates is in situations where the scale parameter is measured in units of time, e.g., "two minutes between events on average", "timescale" seems a not unreasonable term to use, although I must admit I don't believe I've seen it before.

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