If Distribution is Geometric does that mean underlying probability of success for each trial p is fixed?

we know that if p (probability of success at each trial is fixed then the probability of each trial, then probability of first success at kth trial is given by Geometric Distribution

I need to know if other way round is true? i.e. if the distribution (obtained by fitting Geometric distribution on the data) is Geometric then does that mean the probability of success is fixed number p or it might varying one?

• If you write out the formula for the Geometric distribution, how many different values for the probability parameter do you see? Nov 27, 2019 at 4:55
• I understand that there is only one value. However is it safe to assume p is fixed given the distribution is geometric? Nov 27, 2019 at 5:55
• If there is only one value for $p$, how can $p$ be anything other than fixed? That's pretty much the definition of "fixed"... Nov 27, 2019 at 15:19
• Key to formulating an answer is the issue of how you model the variation in $p.$ Are you supposing that it varies (a) deterministically; (b) randomly but independently of other trials; or (c) randomly and perhaps associated with previous trials?
– whuber
Nov 27, 2019 at 17:26

That looks like a profound question to me! My view is that probability distributions are nice bits of math theory, and it's fun to work out the probability of success on trial $$k$$ given a value of $$p$$. To make the math work, you assume things like the observations being independent and identically distributed. I think you are saying you have some real world data? The first thing you are doing is choosing/assuming that the geometric distribution is a plausible model for the variation in the data. But the data are the data. The geometric may or may not be a good model for the data. You hope it's good enough to help you interpret the data. However, as well as choosing the geometric distribution, you pay for that choice with a load of assumptions. Such as the observations being independent and identically distributed. But one massive assumption when using the Geometric is that it is memoryless (it's great fun to prove this). The probability of failure after $$k$$ more events is the same if you have already seen one trial or if you have seen $$n$$ trials. This assumption seems highly implausible for modelling life expectancy say. Can the probability I will die within the next five years really be the same if I am 20 or if I am 110. But ultimately I think that's why your question is brilliant. Probability is great fun as a math exercise. But when you start using it the other way round, to interpret data, you have to make a judgement as to whether things like $$p$$ might indeed be constant.