$$ P(x+y)=P(x)P(y) $$
$$ P(l)=e^{-l/\lambda} $$
💷 where $\lambda$ is the mean free path, it varies based on the properties of the gas; and $l$ is the length before a collision
$$ P^N=\left(1-\frac{\Delta V}{V}\right)^N\approx1-\frac{N\Delta V}{V} $$
$$ \frac{\text{d}P(l)}{\text{d}l}=-\frac{4\pi r^2 N}{V}P(l)=-\frac{1}{\lambda}P(l) $$
💷 where $\lambda=V/4\pi r^2N$
$$ P(l)=e^{-l/\lambda} $$
$$ f(l)=\frac{1}{\lambda}e^{-\frac{l}{\lambda}} \text{ for }l \le 0 $$
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/6308890e-8647-4c5a-8aa3-bf3a432d511d/Hazard_rate.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/6308890e-8647-4c5a-8aa3-bf3a432d511d/Hazard_rate.png" width="40px" /> Hazard rate: the rate of death for an item of a given age (x)
</aside>
<aside> <img src="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/8f73316c-bce7-4216-ba69-e75be429b933/Hazard_function.png" alt="https://s3-us-west-2.amazonaws.com/secure.notion-static.com/8f73316c-bce7-4216-ba69-e75be429b933/Hazard_function.png" width="40px" /> Hazard function: it is defined as
$$ \lambda(t)=\frac{f(t)}{P(t)~} $$
</aside>
$$ \begin{aligned} f(t)\Delta t&=\underbrace{P(t)}_{\text{Pb. of survival until $\small t$}}\times\overbrace{\alpha(t)}^{^\text{pb. of decay in inst. $\small{\Delta t}$}}\Delta t \text{ because: }\lambda=\alpha(t) \\ \frac{\text{d}P}{\text{d}t}&=-f(t)=-\alpha(t)P(t)\quad \lambda=\alpha(t)
\end{aligned} $$
