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### 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 $N(0,1)$...
43k views

### How to generate uniformly distributed points on the surface of the 3-d unit sphere?

I am wondering how to generate uniformly distributed points on the surface of the 3-d unit sphere? Also after generating those points, what is the best way to visualize and check whether they are ...
375 views

### $(2Y-1)\sqrt X\sim\mathcal N(0,1)$ when $X\sim\chi^2_{n-1}$ and $Y\sim\text{Beta}\left(\frac{n}{2}-1,\frac{n}{2}-1\right)$ independently

$X$ and $Y$ are independently distributed random variables where $X\sim\chi^2_{(n-1)}$ and $Y\sim\text{Beta}\left(\frac{n}{2}-1,\frac{n}{2}-1\right)$. What is the distribution of $Z=(2Y-1)\sqrt X$ ? ...
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### Difference between the assumptions underlying a correlation and a regression slope tests of significance

My question grew out of a discussion with @whuber in the comments of a different question. Specifically, @whuber 's comment was as follows: One reason it might surprise you is that the assumptions ...
399 views

### Average absolute value of a coordinate of a random unit vector?

Let $\vec x$ be a random unit vector (that is, a random vector on the unit sphere). Let $x_i$ be the $i$'th coordinate (if it is easier, you can assume we are in 3-dimensional space). What is the ...
3k views

### Expected value of dot product between a random unit vector in $\mathbb{R}^n$ and another given unit vector

I am wondering what is the $\mathbb{E}[(x\cdot v)^2]$ where $x$ is a random unit vector in $\mathbb{R}^n$ and $v$ is a given unit vector in $\mathbb{R}^n$. By $(x\cdot v)$ I mean the dot product ...
1k 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 ...
1k views

### What's the intuition for a Beta Distribution with alpha and / or beta less than 1?

I am curious for myself, but also trying to explain this to others. The beta distribution is often used as a Bayesian conjugate prior for a binomial likelihood. It is often explained with the example ...
216 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 ...
163 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 ...
912 views

### Cosine similarity between a clean signal and its noisy version

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 ...
646 views

### Moment/mgf of cosine of directional vectors?

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 $\mathcal N (0,\Sigma)$, ...
320 views

### Distribution of argmax of Complex Gaussian

The question is about some iid complex Gaussian. Let $\vec{h} = [h_1,\ h_2,\ \ldots ,\ h_n]^T$ where $h_i = x_i + j \ y_i$ and $x_i,y_i \sim \mathcal{N} (0,\sigma^2 = \frac{1}{2})$ I ...
I read in a textbook (Japanese one) that the spherical covariance function is only valid for dimensions $d = 1,$ $2,$ and $3.$ I have the following questions: Does that mean the spherical covariance ...
### Distribution of $X'X$ if $X\in\mathbb{R}^{T \times N}$ and $X_i'\sim N(\mu,\sigma^2I_N)$
Let $x_i\in\mathbb{R}^N$ be multivariate normal distributed with mean vector $\mu\in\mathbb{R}^N$ and no correlation: $x_i\sim N(\mu,\sigma^2 I_N)$. Given $T$ iid samples, define the matrix X:=\left(...