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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?
Jul
29
awarded  Yearling
Jul
21
accepted Proof that the first reappearing remainder when dividing one by a prime number is one
Jul
21
comment Proof that the first reappearing remainder when dividing one by a prime number is one
Thank you! Is it perhaps possible to proof that the first reappearing remainder when dividing one by a prime number is one directly follows from the condition that if the expansion of $1/p$ recurs with period $k$ then $10^k−1$ is divisible by $p$? I don't see that this is a direct consequence of your answer.
Jul
21
revised Proof that the first reappearing remainder when dividing one by a prime number is one
added 256 characters in body
Jul
21
revised Proof that the first reappearing remainder when dividing one by a prime number is one
edited tags
Jul
21
asked Proof that the first reappearing remainder when dividing one by a prime number is one