Generalise exponential growth distribution as a tilted distribution

[SamuelBrand1]

The exponential growth distribution is equivalent to exponentially tilting the $\mathcal{U}(0,1)$ distribution. In general for abs. cont. distributions it changes the pdf from $f(t) \to e^{rt} f(t) / M(r)$ and on the linked wiki page there are some useful distributions where the tilted version is known (e.g. Gamma tilted by $r$ but conditioned onto [0,1] is analytically doable).

[seabbs]

Good idea

[seabbs]

@SamuelBrand1 I think you did a bit more exploration of this. Can you throw it up if so?

[SamuelBrand1]

@SamuelBrand1 I think you did a bit more exploration of this. Can you throw it up if so?

I did a wee thing on if primary time had a reference of (say) truncated Normal on $[0, w_P]$ rather than a uniform then we still get the primary CDF dropping out analytically (because tilted Normal is equivalent to a Normal with shifted params).

The modelling idea here would be combining the concept of certain times in a window being a priori more probable times for a primary event and then combining that with the effect of exponential growth in event incidence.

But I don't know if we want to go forwards with this.

[seabbs]

Yes I think so