Advanced Math/and equation to describe a scenario
QUESTION: I need help with a math equation to describe a scenario.
Here’s the scenario:
Picture a large piece of cheese. There are 100 mice released at the same time that compete for the cheese.
On average each mouse would get 1/100th of the cheese but if you looked at the distribution of mouse performance on a graph it would look like a bell curve with half the mice performing better than average and half performing worse.
If we run this test a hundred times the average mouse will still get 1/100th of the cheese and the distribution of performance will still be a bell curve but the shape of the bell curve will change. If we assume that the performance of each mouse is random during each run (no inherent mouse skill) the bell curve should become more “narrow” but the center of the curve will remain in the same location. Eventually, if we run this to infinity the bell curve should in essence become a vertical line on the graph at 0.01
Any ideas how to describe this in an equation?
ANSWER: A function that fits your scenario would be the generalized Gaussian (bell curve), which, for x = fraction of cheese per mouse, is
f(x) = (1/τ)･exp(-𝛑･x^2/τ^2)
which gets thinner as τ gets smaller. It has the added advantage of having unit area like a probability distribution. The parameter τ plays the role of a standard deviation. The term "generalized" used above is an actual mathematical term and refers to so-called regular sequences which, loosely speaking, converge to a delta function (your infinitely thin vertical line).
I'm not sure I understand your scenario where the bell curve gets thinner as more tests are run. If you are talking about the distribution of the estimate of the mean amount a mouse gets, then yes, the standard deviation of this estimate-of-the-mean goes down as the square root of n = number of tests. If you are talking about generating more members of the ensemble representing the 100-mouse test, then the distribution will still be bell shaped, albeit with better and better estimates of the distribution parameters.
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QUESTION: By "more narrow" I mean that the average mouse would get 0.01 on each run. Let's say the range of performance of the 100 mice on each run is 0.005 to 0.015. If the performance on each subsequent run is truly random (a rat could get 0.005 then 0.013, 0.006, 0.011... on subsequent runs) eventaually the range or the spread of the bell curve will get smaller (more narrow) since every individual mouse will eventually average exactly 0.01 as we approach infinity runs.
Does that make sense? Does this change the equation at all?
Also, in your response, one of the variables came through as a square.
I still think you are talking about the distribution of the mean. However, the equation is still good for that case (or the case you are trying to model).
Is the "square" variable the symbol for pi = 𝛑? If not, if you are still confused by some screwy type-font-whatever, describe it and I'll try to give you the correct symbol.