# Tagged Questions

Questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution.

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### Looking for references related to an inequality in order statistics

I was reading the paper "on the minimum of several random variables". In example 10 item (ii) it states: Let $1\leq k\leq n$. Let $g_i,i\leq n$, be independent $N(0,1)$ Gaussian random variables. ...
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### What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?

If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed. What is the distribution of $Z$ if $X$ and $Y$ are correlated (...
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### Independent normal distributions

I found two theorems with a similar content and want to find out which one is true: Let $X,Y$ be normally distributed random variables and $X+Y$ is also normally distributed or $(X,Y)$ is ...
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### Write $\Phi_n(\sqrt{y-1})$ in terms of $\Phi(y)$ and $n$. ($\Phi_n$ CDF of a $\mathcal{N}(0,\frac{1}{n})$)

I'm trying to solve the following problem: Let $X_n \sim \mathcal{N}(0,\frac{1}{n})$, and let $Y_n$ be the variable defined by: $$Y_n(\omega)=\int_{-1}^1 | X_n(\omega)-t |\,dt$$ Let $F_{Y_n}$ ...
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### Standardized Normal Distribution Problem

Mopeds (small motorcycles with an engine capacity below $50~cm^3$) are very popular in Europe because of their mobility, ease of operation, and low cost. The article “Procedure to Verify the Maximum ...
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### Simplifying covariance matrices in distributions

In the multivariate Gaussian distribution, it is required that the covariance matrix be positive semidefinite. I have read that a positive semidefinite matrix $\Sigma$ can be written as $LL^{T}$. I ...
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### Means and Covariances of powers of a normal distribution

Let $X$ be a normally distributed random variable, with mean $\mu$ and variance $\sigma^2$. Consider a random vector $$V = \left[ X^n, X^{n-1}, \dots, X^2, X, 1 \right]^T$$ What is the expected ...
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### Conditional expectation of independent normals

As it was proved in the answer here, for $Z_1, Z_2$, two independent and identically distributed random variables. Then we have: $$\mathbb E[Z_1\mid Z_1+Z_2] =\frac{Z_1+Z_2}{2}.$$ However, suppose ...
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### $(X_n)_{n\in\mathbb{N}}$ independent with standard Gaussian distribution

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of independent random variables, each with standard Gaussian distribution. For a given $K>0$, prove that: \lim_{n\to\infty} \frac{1}{n}\log{P\left(\...
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### Non-uniform convolution with discrete wavelet transform

I understand that if you have a circular N-dimensional convolution matrix, it can be diagonalized with the Fourier transform of the convolution operator. This makes it easy to calculate the density of ...
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### Expectation of the matrix product $Q A (Q A)^h$

Let $Q$, of dimension $m \times n$, be a matrix with i.i.d. complex Gaussian elements; we suppose that these elements have zero mean and have the same variance $= v$. We define $A$ as an $n \times k$ ...
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### Compound Distribution — Log Normal Distribution with Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (mean of the log of ...
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Consider the two independent random variables $X$ and $Y$ where each random variable is 3-dimensionally normal distributed with $X \sim \mathcal{N}(\mathbf{0},\Sigma_X)$ and $Y \sim \mathcal{N}(\... 0answers 52 views ### Distribution of$aX+bX^2+cX^3$where$X$is standard normal I am looking for some distributional characteristic (for example a characteristic function) of a random variable which is defined as$aX+bX^2+cX^3$, where$X$is a standard normal variable. Is there ... 0answers 49 views ### Understanding the Normal Distribution? If a sample is normal with observations independent and identically distributed:$\mu|\sigma^2 \propto N(\beta \,,\,\sigma^2/\, n_0)$How can I show that$\mu\,|\,x_1,x_2,....x_n\,,\,\sigma^2 \sim ...
Answering a recent question, I realized that the nearest point for a planar Poisson point process (with constant intensity $\lambda>0$) is normally distributed. Indeed, it is easy to see that if \$...