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Flipping the order of Poisson and exponential:

Normal: A great measure for how far off a set of experimental observations might be off from the “true” underlying probability. If you flip a fair coin 10 times odds are you are NOT going to get 5 tails and five heads, but you have the same chance of having “extra” heads or tails, you don’t expect to have ALL heads or all tails. Sure, you could use the binomial, but as the number of trials in a binomial goes to infinity it approximates normal anyway.

Log normal: Take a draw from the normal distribution, then just take to that power. Why do you use it? Well, it’s never negative for one, and by the properties of logarithms you can just sum the normals to get the compounded return (multiplying log normals together), so it’s super popular in finance/

Exponential: This one is trippy because it is a “memoryless” distribution. If I am waiting on a guy to keep over dead, my expectation of the likelihood of that event really hinges on how old he is. If he’s 80 I expect it sooner than if he’s thirty. But if I am waiting for a carbon atom to decay, it doesn’t matter if I know this is a fresh carbon atom that just came out of a decay process or millions of years old, the probability of it decaying in the next stretch of time is the same. The amount of time it takes to have a 50% chance of having decayed is called the half life.

So say the half life 10 minutes. I watch for ten minutes, and it has not decayed. The probability of it decaying in the next 10 minutes is still 50%.

Poisson: How many times are you going to observe an exponentially distributed event in a time frame? So if I am watching my atom for an hour and the half life is still ten minutes, and every time an atom decays I start watching a new one, how many atoms am I likely to look at?