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 Jul 30 revised How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$? added 176 characters in body Jul 30 answered How to simplify a sum with binomial coefficients multiplied by $k^3/2^k$? Jul 5 comment Combinatorics: Mean and Variance of an indicator function of items arranged in a circle. You have hit on the crux of the matter: the question, to be unambiguous, must somehow include enough information to enable calculation of all probabilities. In some settings those probabilities are given explicitly, but usually some particular process whereby the randomization is achieved is described. When the process consists of placing each blue ball within one of the spaces between red balls, all with equal probability (of $1/15$), and doing so in a way that the placement of one blue ball affects no other blue balls, you get the multinomial probabilities. Jul 5 comment Combinatorics: Mean and Variance of an indicator function of items arranged in a circle. Perhaps an illustration would help. In the case $n=k=2$ there are two distinguishable configurations modulo symmetries of the circle: RBRB (red-blue-red-blue in counterclockwise order) and RBBR. In the multinomial scenario both are assumed to have probability $1/2$, while in the permutation scenario they are assumed to have probabilities $1/3$ and $2/3$, respectively. Some people have argued that absent any additional information, one should assume the maximum-entropy distribution. That's the multinomial one in this case--not the permutation one! Jul 5 comment Combinatorics: Mean and Variance of an indicator function of items arranged in a circle. Yes I can still come up with two (and even more) solutions. In this post I have explained two ways of looking at the situation that are perfectly consistent with the problem. If you disagree, the burden is on you to demonstrate how one (or even both!) violate some explicitly-stated criterion in the problem statement. I would even go so far as to accept that there are some conventional implicit criteria in questions like this one: the obvious one is that the distribution ought to be invariant under symmetries of the configuration. That is the case in both scenarios. Jul 5 comment Combinatorics: Mean and Variance of an indicator function of items arranged in a circle. No, that's not right. "Random" is not a quantitative characterization: you have to specify the process involved. The problem statement does not give enough information to "assess" a unique distribution. Both my answers are correct given their assumptions about how the balls are distributed randomly. One of them happens to agree with yours, but that only means it likely adopted the same assumptions you did. Jul 5 revised Combinatorics: Mean and Variance of an indicator function of items arranged in a circle. added 4099 characters in body Jul 3 answered Combinatorics: Mean and Variance of an indicator function of items arranged in a circle. Jul 1 comment How to understand Demartines theorem This is a minor variant of the Central Limit Theorem, which asserts the standardized version of $||X||_2^2$ is asymptotically Normal. It requires that $X$ have finite variance. Jun 26 comment Transforming vector elements to element indices Because matrix operations represent linear transformations and the relationship between Y and X is not linear, no such Q exists. Jun 18 comment Does affine equivariance implies shape unbiasedness? Some things to consider: when $\sigma(X)=0$ (the $p\times p$ zero matrix) the procedure is trivially equivariant but obviously not unbiased. More generally, when $\sigma$ is equivariant unbiased and $p\gt 1$, the procedure $\tau(X)=\lambda\sigma(X)$ is also equivariant but when $\lambda^p\ne \lambda$ it must be biased. This suggests that dividing $\sigma(X)$ by its determinant might not be the right definition of "unbiased" to be using. Apr 7 comment Holder's inequality Your second statement is not well defined until you assume both $X$ and $Y$ are strictly positive. Mar 7 reviewed Approve another question on surds and how to use math symbols in this site Feb 26 comment Lower bound for non-negative definite matrix The matrix $$\left( \begin{array}{cc} 0 & 1 \\ -1 & 2 \\ \end{array} \right)$$ is non-negative definite but your expression equals $-2$. Perhaps you would like to stipulate that $A$ also be symmetric? Feb 16 awarded Good Answer Jan 16 awarded Custodian Jan 16 reviewed Leave Open How do you find the angle between a diagonal of a cube and one of its faces? Jan 16 reviewed Looks OK How do you find the angle between a diagonal of a cube and one of its faces? Jan 16 reviewed Looks OK Integration Techniques - Adding [arbitrary] values to the numerator. Jan 16 reviewed Approve Card game question