1
vote
0answers
15 views

Shortest path length when edge length is limited

$N$ nodes are uniformly distributed in a square whose side length is $1$. There exists an undirected edge between two nodes, if and only if the distance between them is less than or equal to $r$. Here ...
2
votes
1answer
440 views

Expected value of the distance between 2 uniformly distributed points on circle

I have the following problem (related to Bertrand): Given a circle of radius $a=1$. Choose 2 points randomly on the circle circumference. Then connect these points using a line with length $b$. ...
0
votes
0answers
11 views

Probability of error in two dimensional signal space

The likelihood decision rule for two dimensional signal space is $r_2$ > $r_1$ for $H_1$ hypothesis and $r_2$ < $r_1$ for $H_0$ hypothesis. The range of $R=[r_1~r_2]$ is $-\infty<r_1<\infty$ ...
1
vote
2answers
91 views

random circle with radius r on cartesian plane, probability of it not cutting x and y axis with intercepts.

I have a tough question here. Choose a circular disk of radius r on the cartesian plane. What's the probability it is not cut by horizontal lines with integer y intercept, or vertical lines with ...
1
vote
2answers
67 views

Integral of convex set

Let $D$ be a convex set and $X_1,\dots,X_d$ be integrable random variables. If $X= (X_1,\dots,X_d)$ is in $D$ almost surely, why is it true that the vector $a= (\mathbb{E}X_1,\dots,\mathbb{E}X_d)$ ...
5
votes
0answers
165 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 ...