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Jun
15
revised Probability (usage of recursion)
added 325 characters in body
Jun
15
answered Probability (usage of recursion)
Jun
15
comment How to show that $\sum\limits_{k=1}^{n-1}\frac{k!k^{n-k}}{n!}$ is asymptotically $\sqrt{\frac{\pi n}{2}}$?
@PeterR: Often mathematicians will use "elementary" to mean "does not use complex analysis". As you can see, "elementary" does not mean "easy."
Jun
15
comment Conjecture: The following sum is asymptotic to $\sqrt{9πm/8}$
Interesting. This immediately gives that three collisions occur in less than an expected $2 \sqrt{\pi m/2} = \sqrt{2\pi m}$. (In fact, I will guess that it's $15/8 \sqrt{\pi m/2}$.) Since $\sqrt{2\pi m}$ is the square root of the circumference of a circle of radius $m$, there is clearly a geometric proof we're missing. :)
Jun
15
comment How many even number in a sequence are there?
Your iff is false in both directions.
Jun
15
comment Second pair of matching birthdays
@ShreevatsaR: The reason my simulation made me believe the answer was not proportional to $\sqrt{M}$ was that--as your program shows--the multiplier starts above 2.1 and gradually settles to 1.88... But your argument in your answer that it must be a multiple of $\sqrt{M}$ is quite convincing.
Jun
15
revised How many expected people needed until 3 share a birthday?
added 263 characters in body
Jun
15
accepted How many expected people needed until 3 share a birthday?
Jun
15
comment How many expected people needed until 3 share a birthday?
Following the link to your other question, and then to the Sedgewick/Flajolet book, and then a reference from there, I found a paper that gives the derivation for this result: sciencedirect.com/science/article/pii/S0021980067800759
Jun
15
comment How many expected people needed until 3 share a birthday?
Thanks, Byron. I had guessed that $E(T) \approx c M^{2/3}$, but simulations I ran showed $c$ growing slightly with $M$ so I thought the $2/3$ exponent was a tad low. It could instead be the effect of lower-order terms in the asymptotics.
Jun
15
comment How many expected people needed until 3 share a birthday?
I am asking for the expected number of balls where a 3-way collision occurs. But I would be happy to learn the "median" value, which is the number of balls where the a 3-way collision has probability $\approx$ 1/2.
Jun
15
revised How many expected people needed until 3 share a birthday?
deleted 37 characters in body
Jun
15
revised How many expected people needed until 3 share a birthday?
added 1161 characters in body
Jun
14
asked How many expected people needed until 3 share a birthday?
Jun
14
comment Second pair of matching birthdays
I have worked (unsuccessfully) at finding a closed form for the constant $c \approx 1.88$ that you approximate via your python program above. Unfortunately the integral that so nicely turns into $\sqrt{\pi/2}$ ends up being much harder with the ${n \choose 2}$ multiplier.
Jun
14
accepted A seeming paradox in a coin-flipping game
Jun
13
answered Why is it that, $\forall x \in \mathbb{Z},\ x^5 \equiv x \pmod{10}$?
Jun
10
comment Prove that if $n$ is a composite and $p \gt \sqrt[3]n$, then $n/p$ is a prime.
If you look at the edit history, it used to say "composite" at the time my response was offered.
Jun
4
accepted Second pair of matching birthdays
Jun
1
comment Second pair of matching birthdays
Very nice! I have been working on this problem since I posted it and I followed virtually the exact same steps as you do above, but you are faster. I just last night wrote the same program you did (in C instead of Python, but they're almost identical!). I feel guilty seeing all the work you did on this... I'm doing this just for fun (it's summer after all!). Cheers.