Consider an integer cube of size $\sqrt{k} \times \sqrt{k} \times \sqrt{k}$, where $k$ is an asymptotically large perfect square number. Place k points in this cube at uniformly ra …

Two disjoint unit discs $D_1$ and $D_2$. Inside them there are random poisson points with intensity $\lambda$. For a given real $r>0$, what is the probability that there exist a po …

For a reversible Markov chain $X_{t}$ on $\mathbb{R}^{n}$ with transition kernel $K$ and stationary distribution $\pi$, it is well-known that the `spectral gap' (basically, the siz …

Can anyone help with the following (slightly weird) random walk question? I have a random walk starting at $X_0 = 1 = S_0$ where the steps $X_i$ are independent uniform random vari …

Hi, We consider subspaces of $\mathbb{R}^N$. Suppose that we have a property called $\mbox{Prop}$ that apply to subspaces of $\mathbb{R}^N$. That is to say a function from the se …

I want to estimate exponentially the following probability:

Two positive numbers $\alpha$ and $\beta$ are given. We are going to describe a process of choosing a random vector on the unit sphere $S$ in $\mathbb R^3$ (given by $x^2+y^2+z^2=1 …

The density of a wrapped normal distribution is given by $$\frac{1}{\sigma \sqrt{2\pi} }\sum _{k=-\infty }^{\infty }\text{Exp}\left[\frac{-(\theta -\mu -2\pi k)^2}{2\sigma^2}\right …

Hi, Suppose i have n (two dimensional) points randomly chosen on a plane. Both x and y dimensions are intervals [0, 1). What is the expected size of the kth smallest item? Note: …

Suppose I have a probability density function defined on a region in the plane (in my case, the pdf is of the form $f(x) = \alpha\|x\|^{-\beta}$, and the region is the unit disk). …

Covering a circle randomly with arcs has been well studied in the past (Geometric Probability - Solomon). But the problem when the circle is changed to a line segment doesn't seem …

People haven't been rushing in to answer this question I asked on stackexchange yesterday. The way I phrased it initially was this: Does anyone know the probability distribution o …