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Jun
15
revised Probability (usage of recursion)
added 325 characters in body
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