Sometimes I see the exponential distribution defined as follows:
$$f(x) = \lambda e^{-\lambda x}$$ when $x > 0, 0$ otherwise
I have also seen it defined like so:
$$f(x) = \frac{1}{\lambda} e^{-\frac{x}\lambda}$$ when $x > 0, 0$ otherwise
So what do these different ways of defining the same function represent? Say I was dealing with the population mean times between accidents on a road, which one would be more appropriate? Or is whichever one you choose completely down to personal preference?
On
A matter of taste. If $\lambda$ is used as the rate parameter, which is the common usage, then we have $E[X] = \frac{1}{\lambda}$. However, it may sometimes be more intuitive to let the first moment parameterize the distribution (e.g. like the Poisson distribution), so if we have $E[X] = \mu = \frac{1}{\lambda}$, then the PDF would be $f(x) = \frac{1}{\mu} \exp \left(-\frac{x}{\mu}\right) = \lambda \exp \left(-\lambda x\right)$.
They represent the same distribution. It would be better not to reuse $\lambda$, so to say the second is $f(x) = \frac{1}{\kappa} e^{-\frac{x}\kappa}$ Then $\lambda=\frac 1{\kappa}$ converts between them. If $x$ has units, $\lambda$ has the units of $\frac 1x$ while $\kappa$ has the same units as $x$. If you are interested in mean time between accidents and $x$ is time, you need $\kappa$ as that has the correct units. If you are interested in mean number of accidents per year, you need units of inverse time and the $\lambda$ formulation is appropriae.