Michael Lugo
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 Apr18 comment Distribution Question Stat Homework questions should be tagged with the "homework" tag. Apr14 answered Uniform Distributions - Probability Apr12 answered Uniform distribution on a simplex via i.i.d. random variables Apr9 comment Collisions in a sample of uniform distribution These are hard if you've never seen them before, but they're standard problems. Now if the distribution were anything but uniform... Apr8 comment discrete math book suitable for younger person? I don't know of any, but there should be. Apr7 comment Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$ @Fabian: I know that. But that just reduces the question to where one-fourth of my constant comes from. Apr7 comment Looking for a function $f$ such that $f(i)=2(f(i-1)+f(\lceil i/2\rceil))$ I agree that a closed-form solution is unlikely. Numerically one gets $f(i) \sim 6.4319461097 \times 2^i$. This seems like what one would expect but I have no idea where that constant comes from. One strategy for guessing a closed-form solution would be to try to divide out the largest power of 2 dividing each $f(i)$, but even that leaves you with some pretty big numbers. Apr6 comment Sum of series $\sum_{k=1}^\infty{-\frac{1}{2^k-1}}$ Well, yeah, that is naive. Maybe my approach is semi-naive. Apr5 comment Sum of series $\sum_{k=1}^\infty{-\frac{1}{2^k-1}}$ What's the "naive approach" here? To be honest, I would have called the approach you used the naive approach. Apr5 comment Sum of series $\sum_{k=1}^\infty{-\frac{1}{2^k-1}}$ I'm not really a number theorist. But computing $\tau(n)$ in general would be at least as hard as primality testing, since $n$ is prime if and only if $\tau(n) = 2$. Apr5 comment How do we arrive at the conclusion that P(Head) =0.5 for a fair coin? Why am I not surprised that Diaconis has worked on this? Apr5 comment Intermediate Text in Combinatorics? Bona also has a book at a higher level than "A Walk Through Combinatorics", entitled "Introduction to Enumerative Combinatorics". I don't know this book but if it's in your library it's probably worth looking at. Apr5 comment Usefulness of Variance You're right about the dice. Apr5 comment Usefulness of Variance I think it's interesting that we gave essentially the same answer. Apr5 answered Usefulness of Variance Apr5 comment How do we arrive at the conclusion that P(Head) =0.5 for a fair coin? Amit: whether a coin comes up heads depends only on the position and orientation that it has when it leaves your hand and the velocity and angular momentum that you give it. (I may not have the list exactly right, but the point is it's some short list of classical physical quantities.) If we knew these quantities exactly we could tell in advance whether the coin will land heads or tails. Apr5 comment How do we arrive at the conclusion that P(Head) =0.5 for a fair coin? Eric: the Gelman-Nolan article mentions an experiment by Kerrich where a wooden disc was coated on one side with lead; the non-coated side faced down 679 times out of 1000. Apr5 comment How do we arrive at the conclusion that P(Head) =0.5 for a fair coin? Eric: I didn't completely specify the model. In some proportion of coin tosses the coin will be close to vertical when it hits the ground. Then the weight distribution might become relevant. Strictly speaking, my argument only applies to the position of the coin while it's in the air. Apr5 comment How do we arrive at the conclusion that P(Head) =0.5 for a fair coin? Arturo, isn't "how close" supposed to be proportional to $1/\sqrt{n}$? Apr5 answered How do we arrive at the conclusion that P(Head) =0.5 for a fair coin?