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 Oct 7 comment Algorithm for optimizing width length of classes of an ordered list of data points under certain conditions @whuber: Do you think that the missing objective function is the problem? So let's take the "signal-to-noise ratio" as the function to optimize, i.e. the ratio of mean to standard deviation (en.wikipedia.org/wiki/Signal-to-noise_ratio). Oct 6 revised Algorithm for optimizing width length of classes of an ordered list of data points under certain conditions added 89 characters in body; edited title Oct 6 asked Algorithm for optimizing width length of classes of an ordered list of data points under certain conditions Sep 26 accepted How to verify that the average is really a simple calculation? Sep 26 comment How to verify that the average is really a simple calculation? Thank you, joriki. The most enlightening part for me is: "[...] the question is then how to tell whether all the possible matches have the same probability of being drawn in the second draw "even though" two balls have already been removed. This is clearly the case, since the balls that were removed were themselves picked randomly with uniform distribution, so there's nothing there that could break the symmetry between the teams." Sep 25 comment How to verify that the average is really a simple calculation? So if you elaborate on this as an answer this would be great - Thank you again. Sep 25 comment How to verify that the average is really a simple calculation? Yes, you are right! Thank you for clarifying. So, to turn it around: Can you always use the easy way to calculate the average value? Which would mean that there are no situations thinkable where the marginal probabilities change (independent anyway but even when dependent)? Can this be proved? Sep 25 comment How to verify that the average is really a simple calculation? @joriki: Perhaps this one is a little bit contrived but imagine two quantum dice where when one die is thrown the expected value of the other one changes from 3.5 to 4.5. Here they are not only dependent but the marginal probability changes in the experiment so you can't just calculate the overall expected value the easy way. I guess there are better real world examples out there... Sep 25 asked How to verify that the average is really a simple calculation? Sep 25 revised How to calculate the expected value when betting on which pairings will be selected deleted 31 characters in body Sep 25 awarded Popular Question Sep 23 accepted How to calculate the expected value when betting on which pairings will be selected Sep 22 comment How to calculate the expected value when betting on which pairings will be selected Thank you, joriki. Why don't you need conditional probabilities? When a match is set the participating teams are not available for the next draw and the last match will be fixed without even drawing because only two teams will be left. Sep 22 asked How to calculate the expected value when betting on which pairings will be selected Aug 8 answered What do $\pi$ and $e$ stand for in the normal distribution formula? Aug 7 comment What do $\pi$ and $e$ stand for in the normal distribution formula? The plots from my answer would have come in handy ;-) Aug 7 comment What do $\pi$ and $e$ stand for in the normal distribution formula? A further indication that there is no special connection between the two constants via this formula is when you use the general base $a$ but don't square the exponents: Integration won't give you $\pi$ because it is not rotationally symmetric any more (just plot it) - but $e$ still crops up (via the natural log) due to the integration operation. Aug 7 comment What do $\pi$ and $e$ stand for in the normal distribution formula? Yes, this division by the logarithm is due to the horizontal rescaling I mentioned above. That a logarithm shows up is just a consequence of "bringing down" the exponent. The natural logarithm is a convenient closed form. But it is true: $e$ keeps showing up - I have to think about that but at the moment I think that is because of its deep connection to differentiation/integration in general, and that is what we are doing here after all. So I don't think that this special curve really connects $\pi$ and $e$. But always interesting to think about these basic ideas that are so fundamental! Aug 7 comment What do $\pi$ and $e$ stand for in the normal distribution formula? @George: I don't think that Euler's identity is of any help here. See also my answer below. Aug 7 answered What do $\pi$ and $e$ stand for in the normal distribution formula?