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Nov
29
comment Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$
I had this exact proof in mind but thought that actually writing it out would be a chore. Thanks for writing this, so I don't have to.
Nov
22
answered Why do the $n \times n$ non-singular matrices form an “open” set?
Nov
21
answered Elementary formula for permutations?
Nov
18
answered An event has $X\%$ chance of happening. What are the odds of the event happening more/less over time?
Nov
17
comment Course Grade - Statistics Question
You get regression to the mean even in the uncorrelated case, though. In fact you get even more regression to the mean.
Nov
17
answered Probability that a $(n, \frac12)$-binomial random variable is even
Nov
16
comment Course Grade - Statistics Question
I'm sure that the assumption of uncorrelated assessment scores is not valid.
Nov
16
answered Is it possible to balance a chemical equation without using trial and error?
Nov
13
answered Writing combined equation for alternating sequence
Nov
12
answered Which is the “fastest” way to compute $\sum \limits_{i=1}^{10} \frac{10i-5}{2^{i+2}} $?
Nov
11
answered the name of a game
Nov
10
comment What does the exclamation mark do?
"Among the worst of barabarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation n! to signify 1.2.3.(n - 1).n, which gives their pages the appearance of expressing surprise and admiration that 2, 3, 4, &c. should be found in mathematical results." - Augustus de Morgan, 1842.
Nov
7
comment How can we approximate $\sum_{j=0}^n{\sum_{k=0}^j{c^j k^{1/2}}}$ by integrals?
Also, if $0 < c < 1$ is fixed and $n \to \infty$ then that incomplete gamma function should be pretty well approximated by an ordinary gamma function.
Nov
6
comment The probability density function of the ratio of two normal R.V.s
Also, this works whenever the joint density of $X$ and $Y$ is rotationally symmetric around the origin. (If $X$ and $Y$ are independent, though, then that forces $X$ and $Y$ to be mean-zero normal.)
Nov
4
comment Attitude towards exercises in mathematics
Knuth and Stanley both indicate the difficulty of their exercises with numerical rankings, though. I don't have my copy of Stanley with me right now but if I recall correctly he got the idea of numerical rankings for exercises from Knuth.
Nov
3
answered Pronunciation of $M(x)$ and $m(x)$
Nov
3
comment With $n$ balls and $n$ bins, what is the probability that exactly $k$ bins have exactly $1$ ball?
How do you ensure that the $n-k$ balls don't end up with just one of them in a bin?
Nov
3
comment With $n$ balls and $n$ bins, what is the probability that exactly $k$ bins have exactly $1$ ball?
The number of balls in each bin is binomially distributed with parameters $n$ and $1/n$. If we assume that the numbers of balls in each bin are independent, we get your approximation.
Nov
2
comment probablity of random pick up three points inside a regular triangle which form a triangle and contain the center
The circle case is related to an old mathoverflow question of mine: mathoverflow.net/questions/2014/…
Nov
2
comment Evaluating an expression using snake oil
Some people may not know what "Snake Oil" means. See section 4.3 of Wilf, generatingfunctionology: math.upenn.edu/~wilf/DownldGF.html