Generate probability density function from a given relation?

Question

Let's say the life time $\tau$ of a particle to exist with a given velocity $v$ is given by

$\tau \sim \frac{1}{1+v^2}$

The maximum velocity is $U$.

I want to know the probability of finding the particle in velocity range $(0-v)$. I know I need to find the PDF for given relation but I am not sure how to achieve that. The steps I am following are:

1. Find total area under function.
2. Divide function by total area to make it's integral converge to 1.

I don't know how to proceed after this. Any help/hint is highly appreciated.

Answer 1

Assuming that the velocity is uniformly disributed in the interval $[0;u]$ and given the tranformation that maps from the velocity $v$ to the lifetime $\tau$, the pdf of the lifetime results to me

$$f_{T}(t)=\frac{1}{2u t\sqrt{t(1-t)}}\mathbb{1}_{\Big[\frac{1}{1+u^2};1\Big]}(t)$$

This has been obtained using the Fundamental Transformation Theorem (paragraph 2 in the enclosed link)

$$f_T(t)=f_V[g^{-1}(t)]\Big|\frac{d}{dt}g^{-1}(t)\Big|$$

Where

1. It is assumed an Uniform distribution of the velocity over its domain.

$$f_V(v)=\frac{1}{u}\mathbb{1}_{[0;u]}(v)$$

2. The tranformation $T=g(V)$ is the following

$t=\frac{1}{1+v^2}$