Rigon et al 2016, would suggest to define a probability density function as:
- \(p_{_{\ }ex}(t_{ex}|t_{in}=t):=\frac{q\left(t_{in}=t,t_{ex}\right)}{F\left(t_{in}=t\right)}\) representing the probability that a quantity of water entered at time \(t_{in}\) will exit at time \(t_{ex}\)
and relabeling \(t_{ex}\)as \(t\) (and dropping, necessarily, \(t=t_{in}\), letting \(t_{in}\)), it is easy to recognize that this is the so called forward in time distribution (or response time distribution) . Therefore