I read that, due to the memoryless property of exponential distributions, the distribution should be used when the rate of an event is constant during the entire period of time. An example would be the rate of failure for transistors over a number of hours.
But wouldn't this constant rate of an event occurring over time result in a probability density function (PDF) that is a horizontal line? And as such, isn't this incompatible with the exponential PDF desired for exponential distributions?
I'm trying to look at the graphs for exponential distributions (and, thus, their PDF) and reconcile this with the theory I'm reading.
I would greatly appreciate it if people could please take the time to clarify this.
Best Answer
I think the misunderstanding is due to the fact that the term "rate" is often used with different meanings. Sometimes the term "rate" and "probability" are confused, but for me the definition of rate is the one common in survival analysis and it is an instantaneous measure of change. The instantaneous rate $\lambda(t)$ of a a continuos random variable $T$ is defined as. $$\lambda(t) = \lim_{\Delta_t \downarrow 0} \frac{1}{\Delta_t} \mathbb{P}(T\in[t,t+\Delta_t)\mid T \ge t) = \dfrac{f(t)}{S(t)},$$ where $f(t)$ is the density function and $S(t) = \mathbb{P}(T \ge t)$ is the survival function. Therefore in the exponential case you get constant rate: $$\lambda(t) = \frac{\lambda e^{-\lambda t}}{e^{-\lambda t}} = \lambda.$$