Exponential distribution - Using rate parameter $\lambda$ vs $\frac{1}{\lambda}$ 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?

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 $f(x) = \frac{1}{\mu} \exp \left(-\frac{x}{\mu}\right) = \lambda \exp \left(-\lambda x\right)$.
