Unanswered Questions
1,561 questions with no upvoted or accepted answers
12
votes
0
answers
3k
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Fourier transform of a Gaussian process
I would like to discuss and ask a question regarding the Fourier transform of a Gaussian process, if it makes sense.
For that purpose, let me describe the following situation.
Let $z(s)$ be a ...
CommunityBot
- 1
11
votes
0
answers
3k
views
Expected value of softmax transformation of Gaussian random vector
Let $\mathbf w_1,\mathbf w_2,\ldots,\mathbf w_n \in \mathbb R^p$ and $\mathbf v \in \mathbb R^n$ be fixed vectors, and $\mathbf x \sim \mathcal N_p(\boldsymbol{\mu}, \mathbf{\Sigma})$ be an $p$-...
CommunityBot
- 1
8
votes
0
answers
333
views
Simulate Gaussian variables conditional on their sum of squares
Consider a $d$-dimensional Gaussian random vector $\mathbf{Z}=[Z_i]_i$
with mean $\boldsymbol{\mu}$ and covariance matrix
$\boldsymbol{\Sigma}$. What would be the more efficient method(s) to
simulate $...
- 84k
8
votes
0
answers
291
views
Confidence interval for a linear combination of $\mu$ and $\sigma$
Given $n$ independent observations $X_1,X_2,\ldots X_n\sim\mathcal{N}(\mu,\sigma^2)$, where both the variance and the mean are unknown. How can I write down a confidence interval for $\mu+c\sigma$, ...
- 138k
8
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0
answers
1k
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QR decomposition of normally distributed matrices
Assume $M$ is an $N \times k$ Gaussian matrix, i.e., its entries are i.i.d. standard normal random variables, with $N>>k$. Take $D=\text{diag}(\lambda_1, \dotsc ,\lambda_N)$ for some fixed real ...
- 292
8
votes
0
answers
137
views
Can these asymptotic conditional expectations be bounded from above?
Problem Setup
Let $\{X^d_1, X^d_2, \cdots, X^d_n\}$ be a $d-$dimensional zero-mean, i.i.d. random variables. Let $S_n^d$ be
$$
S^d_n = \frac{\sum_{i=1}^n X_i^d}{\sqrt{n}}
$$
Let $Y^d$ be a zero-...
- 6,609
8
votes
0
answers
4k
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Assumptions of correlation test vs regression slope test (significance testing)
If my understanding is correct, then
the test on a regression slope in a simple bivariate regression - i.e. the test of $\mathcal{H}_0$: $b = 0$ in $Y' = a + bX$
and
the test of a correlation, i.e. $\...
- 70.4k
7
votes
1
answer
1k
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Conditional expectation in the multivariate normal distribution
Suppose $(X_1, X_2, X_3)^T$ is multivariate normal.
What is the conditional expectation $E(X_1 \mid X_2, I(X_3 > 0))$?
Here, $I(X_3>0)$ is the random variable that takes the value one when $...
CommunityBot
- 1
7
votes
0
answers
1k
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Square roots of sums absolute values of i.i.d. random variables with zero mean
In an earlier question, I asked about the limiting distribution of the square root of the absolute value of the sum of $n$ i.i.d. random variables each with finite non-zero mean $\mu$ and variance $\...
- 43.5k
7
votes
0
answers
259
views
why use diagonal $\Sigma$ when working with Bayes decision theory?
My prof. said in the class that for Bayes decision rule, the likelihood is Gaussian and in practice, we will almost always work with a diagonal $\Sigma$. Why is that? I know that a diagonal $\Sigma$ ...
- 94.9k
7
votes
0
answers
261
views
How to derive the characteristic function of a polar coordinates representation of a bivariate normal
Suppose to have a bivariate normal variable $\mathbf{x}=(x_1,x_2)$ with mean $\mu$ and covariance matrix $\Sigma$. I move from $\mathbf{x}$ to $(\theta,r)$ where $x_1 = r \cos \theta$ and $x_2 = r \...
- 84k
7
votes
0
answers
2k
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How to compare models with different distributional assumptions for response variable in GLM?
Let's say I have measurements $Y$ which are all positive, and the distribution seems to be somewhat skewed. I'm modelling $Y$ in GLM framework. Now I could set my GLM using different distributional ...
- 136
6
votes
0
answers
125
views
Intuition/meaning of information geometry distances and geodesics?
In information geometry, we consider a manifold of probability distributions, together with the Fisher Information metric (given by the Fisher Information matrix).
I have some intuition (see ...
- 12.3k
6
votes
0
answers
229
views
Why use the student's t-test rather than z-score?
Suppose we are given IID r.v's $X_1, \ldots, X_n$ that are not necessarily normally distributed. Mean $\mu$ and standard deviation $\sigma$ are unknown and we want to construct a confidence interval ...
- 59
6
votes
0
answers
3k
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Distribution of multivariate "$Z$-score"?
Suppose $\mathbf{X}_1, \dots, \mathbf{X}_n \sim N_p(\mathbf{\mu}, \Sigma)$ where $\mu \in \mathbb{R}^p$ and $\Sigma$ is a $p \times p$ covariance matrix.
Suppose $\hat{\Sigma}$ is the sample ...
- 16.1k