Exponential distribution

As an example, let us take $y(x)=-\ln{(x)}$ with $P(x)$ representing a uniform distribution in the interval $[0,1]$. Then

$$P(y)=\frac{dx}{dy}=e^{-y},$$

which is distributed exponentially. This distribution occurs frequantly in real problems such as the readioactive decay of nuclei. You can also see that the quantity $y/\lambda$ has the distribution $\lambda e^{-\lambda y}$.