Probabilites of random geometric objects having certain properties (enclosing the origin, having an acute angle, being convex, ...); expected counts, areas, ... of random geometric objects.
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34 views
Probability of finding a point on or in an $n$-dimensional unit sphere
If a point is chosen at random in an $N$-dimensional unit sphere, what is the probability of falling inside the sphere of radius $0.99999999$? What if $N=3$, $N=10^{23}$, or $N = \infty $?
Okay, that ...
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vote
1answer
42 views
A modified Buffon's needle
A needle 2.5cm long is dropped on a piece of paper that has a very fine parallel lines 2.25cm apart drawn on it.
What is the probability that the needle lies between the two lines?
I can see how ...
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votes
1answer
14 views
craps game odd with a pair of dice
in a dice game craps, Alex rolls a pair of fair dice. if he gets 7 on the first roll, he wins immediately. if the result is any number other than 7, he keeps rolling the dice until he gets that number ...
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votes
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64 views
A lawn, a flower, a pipe and the neighbours
You have a square lawn and a precious flower in the centre. You want to make sure you water the flower, and you don't particularly care how much of the lawn you water. To please your aleatory ...
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77 views
Clarification in a paper
This is regarding a clarification in page 384 of a paper published in Annals of Statistics by Amari.
In page no. 384, he defines $$R_i(t)=\frac{\partial}{\partial \theta_i} ...
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votes
0answers
123 views
Expected Number of Convex Layers and the expected size of a layer for different distributions
It is well-known that the expected number of vertices on the convex hull of random set of points in the plane distributed uniformly within a $k$-gon is $O(k\log n)$ and within a smooth shape (e.g. a ...
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votes
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88 views
Sufficient Statistic for a Geometric R.V.
I have a problem that I know I am very close to the solution for, but I think I just need some more formatting to make it a really clean proof.
The problem goes like this:
Suppose X is a discrete ...
2
votes
0answers
89 views
Integral Geometry Reference Request
I am looking for a good introductory reference (book, lecture notes, survey article) on integral geometry. I am especially interested in the Crofton formula in $\mathbb{R}^n$ and its extensions to ...
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65 views
Spatial distribution of bees
* Please please help! I still get stuck.
We have a forest for bees, consisting of $4$ non-overlapping regions. $80\%$ of the bees seek honey in the forest while $20\%$ of the bees do so outside the ...
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71 views
geometric sum with probabilities
I have the following sum to consider:
$$
p_i/(1+r)^i
$$
for i = 1 to 10
My question is, whether this can be somehow evaluated using some sort of mathematical insight without explicitly computing ...
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votes
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61 views
Bernoulli Trials with two random variables - Run of successes and faliures
Consider a sequence of Bernoulli trials with probability of success $p$. Suppose you started the game with a run of succsses followed by the run of faliures (note that you can learn an unlucky run is ...
