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 Nov 21 comment The Light beam Problem. I think where it says "randomly" you mean "arbitarily"? Nov 20 awarded Enlightened Nov 20 awarded Nice Answer Nov 13 comment PMF table probabilities The question is highly confusing in its present form -- one has to read the comments in order to understand that you answered your own question. Please mark this clearly so that the question makes sense by itself. Nov 12 comment Sample from distribution taking spherical statespace That makes no sense. The density function for a two-dimensional manifold must be a function of two variables. It may not depend on one of them, but unless you specify the second variable, you haven't defined a distribution. E.g., a density $\rho(\phi,\theta)\mathrm d\phi\mathrm d\theta=f(\phi)\mathrm d\phi\mathrm d\theta$ would specify a different distribution than a density $\rho(\phi,\sin\theta)\mathrm d\phi\mathrm d\sin\theta=f(\phi)\mathrm d\phi\mathrm d\sin\theta$. Probably you meant the former? Nov 10 answered Expand generating function $\frac{\exp{\frac{z}{1-z}}}{1-z}$ Nov 10 comment Expand generating function $\frac{\exp{\frac{z}{1-z}}}{1-z}$ You used $[z^n]$ for the coefficients of the exponential generating function at the top but for those of the ordinary generating function at the bottom, so the coefficients differ by a factor of $n!$. (Conventionally that notation is used for the coefficients of the ordinary generating function.) Note that what remains of the expression you're trying to obtain if you divide it by $n!$ is $\sum_{i=0}^n\frac1{i!}\binom ni$ (no square). Nov 10 comment Probability of buses passing along the street. @calculus: It gives the same result (as it should). (I hadn't checked it because it seemed more complicated to me and you hadn't written it out in a form that I could simply paste into Wolfram|Alpha :-) Nov 10 comment Expand generating function $\frac{\exp{\frac{z}{1-z}}}{1-z}$ Here's the OEIS sequence with lots of references. Nov 10 comment Probability of buses passing along the street. You probably intended to imply that the order of the $60$ buses is distributed uniformly over all possibly orders? In that case Mroog's answer seems to be correct, and it yields approximately $0.3671$, so there seems to be an error either in the answer you were given or in Mroog's and my understanding of the question. Nov 10 revised What is the name of this binary operation on vectors? edited tags Nov 7 comment generating function from recurrence relation @user141834: I'm not sure which term you're talking about. There's only one $S(x)$ term, and it's at the beginning. Or do you mean the expression for $S(x)$ in the last equation? (That simply results from solving the preceding equation for $S(x)$.) Nov 6 awarded Nice Answer Nov 2 comment The dimension of the real solution space of differential equation. @MaryStar: I think you'll find the answer to your question here: mathoverflow.net/questions/44853/… (where "degree" is used synonymously with "order"): The solutions of a homogeneous linear differential equation of order $n$ always form a vector space of dimension $n$. Nov 2 comment The dimension of the real solution space of differential equation. @MaryStar: That's quite a general question. Generally the set of solutions of a differential equation need not be a vector space or even a manifold (and thus in particular need not have a dimension). Are you perhaps thinking of a particular type of differential equations? Perhaps linear differential equations with constant coefficients, such as in this question? Nov 1 awarded Popular Question Oct 31 comment Expected number of steps till a random walk hits a or -b. @vonbrand: Quite right, thanks, I've fixed it. Oct 31 revised Expected number of steps till a random walk hits a or -b. correction in response to comment Oct 31 comment Probability of finding a lost ball @kjanko: I don't see why that way of appliying Bayes' theorem would seem preferable to you to how I applied it; perpaps that's a matter of subjective preference. Do we at least agree that the result is the same and both derivations are correct? Oct 30 awarded Nice Answer