How is the randomness of single atom radioactive decay verified experimentally? What experimental setups and statistical tests are used to empirically verify that

*

*the decay of a single atom cannot be predicted, it is truly random


*the decay of a single atom is not influenced by its environment


*the probability of decay is constant over time, and independent of the previous interactions of the atom


*the decay probability is the same for the atoms of the same element isotope
?
 A: Over the last hundred years, many samples of radioactive isotopes have been monitored over a period of time.  They all show an exponential rate of decay.  This indicates that a fixed faction of the atoms in the sample will decay in a given length of time, which can only result from a fixed probability that any one atom will decay within that time.  It seems unlikely that the turmoil within a nucleus would be effected by its environment or history.
A: Let me describe an experiment I did in grad school (not so recent anymore) to detect the decay of cosmic ray muons. Here is a detailed writeup of a very similar experiment to the one I did. Even though the experiment involves muons, the answer I'll give doesn't depend on this detail, except in some details of the experimental set up. The principles of quantum mechanics don't depend on the specific particle you are using. If you don't buy that, you could imagine doing variants of the procedure below with different kinds of decaying particles, and you would always find the same answers to your questions.
The procedure is that we had a box to catch muons. The box was made of a scintillator crystal (material that emits flashes of light when excited), along with photomultiplier tubes attached to the top and bottom of the box. The photomultiplier tubes measure flashes of light (a) when a muon crosses the crystal (ie when it enters or leaves the box) or (b) when a muon decay product (electron) hits the crystal. There are also random flashes due to processes unrelated to muons.
There's some logic that you need to implement to determine which the signals that you detect come from a muon that decayed, as opposed to muons that pass through the detector, or get captured by a nucleus, or are due to events that are unrelated to cosmic rays. It's not too hard to do this, and the details are in the pdf I linked above.
Once you determine which events are muons that entered the box and then decay, our data set consists of two things:

*

*The time at which the muon entered the box

*The time at which the muon decayed

Subtracting those two, we get the decay time for each individual muon.
While the point of the lab was to determine the lifetime (roughly speaking by computing the average muon decay time), we could have used the data to answer your questions, like so:

the decay of a single atom cannot be predicted, it is truly random

If you look at the decay times of the muons in sequence, there is no discernible pattern. You can also check that there are no correlations between decay times (eg: there is no correlation between the value of the 1st observed decay time and the 4th). You can even check that the sequence you get is consistent with Poisson statistics, and that if you repeat the experiment $N$ times each with $M$ muon events, the distribution of sequences you get will be consistent with $N$ draws of sequences with length $M$ from a Poisson distribution.
You can try inventing algorithms to predict the next muon decay time, given all the muon decay times you have seen up until that point. As far as anyone has been able to tell after around one hundred years of quantum theory (depending on how you count), and as quantum mechanics firmly predicts, there is no way you can use any information to predict any individual muon lifetime.

the decay of a single atom is not influenced by its environment

You can repeat the experiment by changing details of the setup, like the room you are in, the scintillator crystal, the material making up the box, changing the temperature of the box, etc. You will find the value you get for the lifetime doesn't change.

the probability of decay is constant over time, and independent of the previous interactions of the atom

You can repeat this experiment multiple times at different times of day, different times of years, different years, etc, and you will find the same average decay time for the muon. In fact you can view the fact that there is a "right" answer you are supposed to get in this lab, that I got, and that decades of "children" before me also got, as evidence that the lifetime is not changing over time.

the decay probability is the same for all atoms of the same element

Since the muons in the experiment are always different (since they emerge from different cosmic rays), repeating the experiment multiple times will also test the results with different muons.
