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Say I live in a place that has 40 deaths per billion km driven.

Also, that I drive 30 km a day, 5 times a week, for (53-8) weeks

I am thinking of modeling the number of deaths as a poisson, and therefore my own probability of death as an exponential.

The parameter of a poisson is easy to obtain, given the means: its just 40/(10**9)

So I calculated in scypi

>>> from scipy.stats import expon
>>> p_lambda = 40/10**9
>>> k=30*5*(53-8)
>>> expon.cdf(k,0,1/p_lambda)
0.00026996355328027865

And to be sure I did the same in R

> kms = 30*5*(53-8)
> p_lambda = 40/(10**9)
> a = pexp(kms,p_lambda)
> a
[1] 0.0002699636

I am somewhat inexperienced, so I'd ask for help validating my modelling and my code. Did I miss anything? Do my assumptions make sense?

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    $\begingroup$ What's the matter with estimating your chance as (30 km/day)(5 days/week)(45 weeks)(4E-8 deaths/km) = 0.00027? $\endgroup$
    – whuber
    Commented Sep 20, 2020 at 16:15
  • $\begingroup$ Very glad it is similar. I just thought this would be a better model, but you just proved me wrong $\endgroup$
    – josinalvo
    Commented Sep 20, 2020 at 17:07
  • $\begingroup$ As a practical matter, I would look into how many of those 40 deaths where drunk drivers killing themselves --- or other reckless behaviour killing themselves ... $\endgroup$ Commented Sep 20, 2020 at 22:17

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Since the "number of deaths" is distributed as Poisson, it is the time until the first death that is distributed exponentially. Anyway, your calculations are correct: this is the probability that it will take more than 53-8 weeks for a death to occur.

Also, it is no coincidence that just multiplying the Poisson rate ($\lambda$) by the distance you traveled ($d$) gives a very good approximation to this probability, as noted by Whuber. The probability you calculated is $p=1-e^{-\lambda d}$, which can be approximated by $\lambda d$ when $\lambda d$ is small enough.

When $\lambda d$ is not very small, the approximation breaks: then your calculations still work, but simply multiplying the Poisson rate by the distance does not.

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