>>8165469

>>8165630

The multiplication is supposed to be over the index i (not n).

For each i, we have that [math] p_i(t) [/math] is a function and I take it to be smooth.

Let's be more specific, although these are not fixed rules:

[math] p_i(t) [/math] should to be a function that is zero until a time [math] t=T_i^{introduction} [/math] and grows to some value in [math](0,1)[/math], e.g. 0.2

Background:

There are driver assistance systems [math]i[/math] (like Automatic parking for cars) that that ought to reduce the number of car accidents.

https://en.wikipedia.org/wiki/Advanced_driver_assistance_systems

People in this industry assign pretty random numbers, called potentials [math]p_i[/math], to those systems that ought to capture their effectiveness in doing so, and thus their value. Not all cars will have all systems implemented, and that's why one must make a choice/subset of the available ones.

For a fixed time span, say a month, the idea is that if there wasn't the system i, then there would be [math]n_z[/math] accidents, and with the system, [math]n_i[/math] (with [math]n_z < n_i[/math] because lives are saved etc.) of those would be prevented.

The basic deal is to define

[math] p_i = \dfrac{n_i}{n_z} [/math]

If e.g. [math] p_i = \dfrac{1}{10} [/math] or [math] n_i = n_z/10 [/math], then it means the system i prevents 1/10's of all the accidents.

What's empirically available is the number [math] n_r [/math] of real accidents that happened.

I have the police data for Austria, there are about 5 accidents per hour recorded or whatever.

We don't know [math] n_z [/math], the number of accidents in the world without i's.

For the prevented accidents, we should have

[math] n_i = n_z - n_r [/math].

So

[math] p_i = 1 - \dfrac{n_r}{n_z} [/math]

[math] 1 - p_i = \dfrac{n_r}{n_z} [/math]

[math] n_z = \dfrac{n_r}{1 - p_i} [/math]

for i.

cont.