You are a dungeonmaster hosting Dungeons & Dragons and a player casts 'Spell of Eldritch Chaotic Weather (SECW). You've never heard of this spell before, but it turns out it is quite involved. ...

It's misleading to simply say that statistics is simply the inverse of probability. Yes, statistical questions are questions of inverse probability, but they are ill-posed inverse problems, and this ...

Back in the day, people used logarithm tables to multiply numbers faster. Why is this? Logarithms convert multiplication to addition, since $\log(ab) = \log(a) + \log(b)$. So in order to multiply ...

This happens often in statistics: there are a variety of methods which could be applied in your situation, and you don't know which one to choose. You should base your decision the pros and cons of ...

Here I explain why the asymptotic variance of the maximum likelihood estimator is the Cramer-Rao lower bound. Hopefully this will provide some insight as to the relevance of the Fisher information. ...

If you are wondering why all this machinery is used when something much simpler could suffice--you are right, for most common situations. However, the measure-theoretic version of probability was ...

Conduct an experiment. Randomly downvote half of the new posts at a particular time every day.

The Bayesian interpretation of probability suffices for practical purposes. But even given a Bayesian interpretation of probability, there is more to statistics than probability, because the ...

As long as the sponsors of the site are committed to keeping the site running, it would be premature to declare it 'dead.' It is not out of the question that StatProb.com may experience a revival in ...

I worked out the details omitted from JMS's answer. Let $P_0 = \Sigma_0^{-1}$ and $P_1 = \Sigma_1^{-1}$ We have $$KL(N_0||N_1) = \int N_0(x) \frac{1}{2}[\ln |P_0| - \ln|P_1| + (x-\mu_0)^T P_0 (x-\... View answer Accepted answer 6 votes A rough answer is that the MLE of a location parameter of distribution with non-translation-invariant support can converge faster than O(1/\sqrt{n}). In this case, the divergent entry in the ... View answer 6 votes EXAMPLE: say there are three sites, and we want to compare the well-roundedness of the Users A, B, C. We write the reputations of the users across the three sites in vector form: User A: [23, 23, ... View answer Accepted answer 6 votes If you have a linear rod, the center of gravity is the first moment (the expected value), and the moment of rotational inertia about the center of gravity is the variance. (A rod with centrally ... View answer 6 votes First, here are some commonly known facts which will be useful. Suppose i.i.d X_1, \cdots, X_n have the cumulative distribution function F(X) = P[X \leq x], then cumulative distribution function ... View answer Accepted answer 5 votes Try a location shift on the Gamma(2,1) EDIT: Illustration View answer 5 votes The reason why you are conducting this test is to determine which policy is more valuable, and if value is measured in profitability, then it makes no sense to do statistical testing on any other ... View answer 5 votes I am not exactly sure what your code is trying to do, but it seems like you should be using rnorm() (with a large n) instead of dnorm(), since the functions mean(), var(), etc. are designed ... View answer Accepted answer 4 votes Persi Diaconis wrote an article on "Mathematical Statistics" in the Princeton Companion of Mathematics. He discusses several ongoing research areas in mathematical statistics, including the search ... View answer Accepted answer 4 votes There is a simple interpretation of the above grouping property. Suppose your alphabet is A, B, C,... where the letters have frequency p_1, p_2, p_3, .. Now let S be a random sequence of large ... View answer 4 votes Queue models depend on a few key distributions: the distribution of the time gaps between incoming jobs, the distribution of service times (how long it takes to process a job). Some commonly used ... View answer Accepted answer 4 votes Supposing for independent X, Y, we know E[X], E[XY], var(X), var(XY). Thus we know E[Y] = E[XY]/E[X], E[X^2]=var(X) - E[X]^2 and E[(XY)^2]=var[XY]-E[XY]^2. Since E[(XY)^2] = E[X^2 ... View answer 3 votes An alternative parameterization for the covariance matrix is in terms of eigenvalues \lambda_1,...,\lambda_p and p(p-1)/2 "Givens" angles \theta_ij. That is, we can write$$\Sigma = G^T \...

To add to onestop's response, it was confirmed on math.SE that the polynomials $$w_1 = x_1 + \cdots + x_n$$ $$w_2 = x_1^2 + \cdots + x_n^2$$ $$\cdots$$ $$w_n = x_1^n + \cdots + x_n ^n$$ give you all ...

If you weight your measurements (proportion of subpopulation/proportion of subpopulation in sample), your estimates will be unbiased. I assume this is what you meant by "poll results being ...

There is no simple answer for how to "establish a relationship between two variables;" indeed, your question is one of the central issues in statistics and research is still going on on how to do this....

EDIT 2: We have to simplify the problem by removing the restriction that $x \ne y$ always in case two. Otherwise correlation issues vastly complicate the answer. Let $|v|$, the 1-norm, denote the ...

Suppose you have Case 1: A=200, B=100 C=100, D=200 versus Case 2: A=200, B=0 C=200, D=200 The B=0 in case 2 means that case 2 provides much stronger evidence than case 1 of a relationship between ...

EDIT: After some reflection, I modified my answer substantially. The best thing to do would be to try to find a reasonable model for your data (for example, by using multiple linear regression). If ...