30 views

### Variance of Guassian Products [duplicate]

Suppose I have to vectors $w$ and $x$, each of size $[512,1]$. Each element of $w$ and $x$ is an i.i.d sample from a Guassian Distibution with mean 0 and variance 1. So $x_i$ and $w_i$ follow $N(0,1)$...
28 views

### Why is the marginal pdf of $x$ constant? $x$ is the coordinate of the points uniformly distributed on the surface of a sphere [duplicate]

Consider a sphere of radius $R$ centered at the origin with points uniformly distributed on the surface. What is the marginal pdf of the x-coordinate of these points? Apparently the answer is  f_X(x)...
21 views

2k views

### Why are points uniformly distributed on a sphere in 3D uniformly distributed in component coordinates?

I've generated uniformly random points on a sphere (in 3D). As expected, all azimuthal angles are drawn with equal probability and it's less likely to draw points close to the poles: However, when I ...
231 views

### Null distribution of subspaces similarity, or what is the distribution of $\mathrm{tr}(AA'BB')$?

What is the distribution of $\mathrm{tr}(AA'BB')$ where $A$ and $B$ are two random matrices of $d \times k$ size with orthonormal columns? Maybe the expected value is easier to compute? A fallback ...
394 views

### How to sample uniformly points around a neighborhood of a point lying on a n-sphere?

Given a point $x$ lying on the surface of a n-sphere $S$, what is an efficient way of randomly sampling points $x_k \in S$ such that their distance from $x$ is at most $r$? ($\|x-x_k\| < r$) Can ...
Can anybody suggest how I can compute the second moment (or the whole moment generating function) of the cosine of two gaussian random vectors $x,y$, each distributed as $\mathcal N (0,\Sigma)$, ...
Given a $D$-dimensional datum that is an iid sample from a spherical Gaussian distribution, and the noise-corrupted version of that datum generated by adding spherical Gaussian noise, is there a ...