I'm recently trying to understand where the exponential distribution is originating from, and while learning the exponential distribution, I started exploring whether or not a similar alternative distribution would have some practicality in real world practice.
The exponential distribution pdf is $f(x) = k\mathrm{e}^{-kx}$, and its cdf on $[0, \infty)$ is equal to $1$ for any $k$.
The distribution I came up with has a pdf of $f(x) = k(x+1)^{-k-1}$, and its cdf on $[0, \infty)$ is $1$ for any $k$.
Both functions are decreasing to represent a decreasing pdf at greater values of $x$, but what are some reasons why people would prefer an exponential model over a power model?
Exponential distribution has some very nice property, like memoryless property, relation with Poisson process inter-arrival time, etc, and the distribution itself is relatively simple. It can be used as a toy model to understand the problem.
https://en.wikipedia.org/wiki/Lomax_distribution Whereas the Pareto distribution (Power Law distribution) in general has a heavy-tailed, which used to model those extreme event, and is useful among those, say insurance problem. Many real-world data are actually heavy-tailed and using light-tailed distribution for modelling will create a modelling risk as it cannot reflect the tail behavior. You may also take a look at the term "Black Swan".