Questions tagged [normal-distribution]
This tag is for questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution. The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.
8,335 questions
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Some examples of distributions of real random variables ( Achim Klenke, Probability Theory )
I am reading the Achim Klenke's Probability Theory, A comprehensive course, Example 1.105 and some confution arises in trying to understand some definitions.
Definition 1.103 ( Distributions) Let $(\...
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How to use the Poincaré inequality to bound the variance of the maximum of 𝑛 iid Gaussian random variables
Let $Z$ be the maximum of $n$ independent identically distributed Gaussian variables $X_1,\ldots,X_n$. The cdf $\Phi$ of $Z$ is
$\Phi(x) = F^n(x) $
where $F$ is the cdf of the normal distribution. ...
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Sign of a Gaussian expectation
Let $X$ be a Gaussian random variable with mean=variance=$g>0$, namely its probability density is $p(x)=\frac{1}{\sqrt{2\pi g}}\,e^{-\frac{(x-g)^2}{2g}}$.
I am looking for proofs of the following ...
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The quantile transformation in normal distribution
I'm reading the textbook, Probability and Statistical Inference by Hogg and others.
When explaining the Q-Q plot, it is said that
when $q_p$ is a quantile of the normal distribution N($\mu, \sigma$), ...
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Can a rectangular lattice Gabor system using a Gaussian window-function ever be a tight frame?
This answer appears to claim, that a Gabor frame consisting of coherent states, which from my understanding means a rectangular-lattice based Gabor-frame with a Gaussian window-function, is a tight ...
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Mahalanobis distance with infinite matrices
Let $X\left( t \right)$ be a continuous gaussian stochastic process with zero mean $E\left[ X\left( t \right) \right]= 0$, the index set $ \left[ 0,T \right]$, $t\in \left[ 0,T \right]$. One chooses ...
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Why does the Brownian Motion not follow the Law of Large Numbers?
I am studying physics in third semester, and when I learned about the Brownian motion, I stumbled upon this counter-intuitive conclusion. Let me elaborate:
Assume Brownian motion in 1D, which can be ...
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Tail bound of Gaussian likelihood ratio
Consider a hypothesis testing problem where X,Y are both $N(0,1)$ variables and under $H_0$, X,Y are independent and under $H_1$ (X,Y) follows a bivariate Gaussian with correlation coefficient $\rho$. ...
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Aalternative to the Normal-Inverse-Wishart prior for conditional Gaussian models with Gaussian evidence marginal
I'm working with conditional Gaussian models and looking for an alternative to the Normal-Inverse-Wishart (NIW) prior. I’ll describe the setup and then outline what I’m trying to achieve.
Background
...
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Where in the world is this pmf?
Call this pmf the $\mathrm{BinGIG}(a,b,c,d)$ as it is the compound distribution of a binomial and GIG:
$$
\mathbb{P}(N=k)
=\frac{\binom{a}{k}}{K_{c}\!\big(2\sqrt{db}\big)}
\sum_{j=0}^{a-k}\binom{a-k}{...
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Jensen's inequality vs concavity of $\log(f(x))$: why EM lower bound works for GMM?
I have a doubt about the relationship between Jensen's inequality and the concavity of a composite function, specifically $\log(f(x))$.
Let $f: \mathbb{R}^n \to \mathbb{R}_{>0}$ be a positive ...
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Kahane's inequality and the significance of sub-gaussian growth
I've come across the following Inequality in my reading.
Let $X, Y \in \mathbb{R}^n$ be centered Gaussian vectors, and $f \in C^2(\mathbb{R}^n)$ a function whose second derivatives have sub-Gaussian ...
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Integrating with gaussians emerging from exact solution to Burgers' equation
I'm trying to find exact analytical solutions to Burgers’ equation,
$
\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = \mu \frac{\partial^2 u}{\partial x^2}, \quad \mu>0.
\tag1
$
I'...
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How to calculate $P(X+Y<0 | X<0)$?
For $X \sim N(\mu_x,\sigma_x^2)$ and $Y\sim N(\mu_y,\sigma_y^2)$ which are independent, can I calculate $P(X+Y<0 | X<0)$?
I found this question for independent standard r.v and this one for ...
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Equal-weight averaging sets (designs) for the Gaussian
tl;dr: Are there $t$ points $x_1,\dots,x_t \in \mathbb{R}$ such that
$$ \frac{1}{t} \sum_{i=1}^t x_i^k = \int_{-\infty}^\infty x^k\ d\rho(x)$$
for every $1 \leq k \leq t$, where $d\rho$ is the ...
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