Distribution of the time delay between vaccination and death So please first let me state that I'm absolutely pro vaccine and not trying to demonstrate anything here, I just thought that the VAERS (Vaccine Adverse Event Report System) data was an interesting place to learn some data analysis and data investigation skills on my free time during holidays.
Here is the claim I'm trying to investigate : among the 8k+ deaths linked to the Pfizer vaccine in the US, only a handful have been causally attributed to the vaccine, and the rest is supposed to be temporal coincidences (people will sometimes die, wether they have taken vaccine or not).
To investigate this claim, I propose to look at the distribution of the number of days elapsed between taking the vaccine and dying, for each case recorded in VAERS in 2021.
My hypothesis is : if the death and the vaccine aren't causally linked, the distribution of the observed time delay between vaccination and death should be approximately uniform (people have the same probability of dying the day when they are vaccinated, the morrow, the day after that, the 30th day after that...).
Here is my procedure : using pandas on the 2021 data downloadable from VAERS, I found

*

*8500 death events with non-NaN entries


*8137 remained after removing badly formatted DATEDIED and VAX_DATE entries


*8122 remained after removing entries with negative number of days elapsed between death and vaccine
I plotted two histograms, one with the X axis ranging from 0 to 400 days, the other from 0 to 40 days, see attached.
Contrary to my hypothesis, the distribution is not at all uniform, but instead seems to be exponentially decreasing with a maximum at 0. Also, there is a small bump centered around ~180 days.
My tentavive explanations are :

*

*Not everyone has been vaccinated at the same time and this will tend to skew the distribution to the left, simply because people vaccinated during last month cannot contribute to values greater than 30 days, for example. I dont know how to estimate this effect overall, but since vaccination rates have plateaued for at least a few months, I dont expect it to be significant in the range ~0-100 days. Am I right ? Is there any other probabilistic effect that I'm missing ?


*The adverse events are under-reported, and the older the vaccination event is, the less probable it is that a death is reported in VAERS. This is, I think, almost surely happening, but I wouldnt expect such a big drop in reports between day 1 and 2, especially for an adverse event as tragic as a death.


*The bump at 180 days may be due to the period around april where vaccination rates were steeply increasing.


*The causal link between vaccines and death are stronger than expected, causing the left skew.
Here are the histograms
0-400 days https://ibb.co/YBT7CpG
0-40 days https://ibb.co/4Y1sN42

Your uniformity  premise is flawed.
You said: "(people have the same probability of dying the day when they are vaccinated, the morrow, the day after that, the 30th day after that...)"
Presumably you instead mean the same conditional probability of dying. That is "given they survive to day $t-1$, the probability that they die on day $t$ is constant for each $t$". This (along with independence) then implies a geometric distribution of times (time to first failure in a sequence of i.i.d. Bernoulli trials), not a uniform. ... so exactly the "exponential" you say you observed.
There's nothing to explain, you pretty much see what you should have expected once you express it correctly.
The bump you mention does require some explanation, though. I think that has to do with several of the assumptions not actually being correct (like waning immunity followed by a decrease - perhaps with second vaccinations, boosters, etc, or simply that there was a spike in cases at some point where the timing happened to work that way.)
