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

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10
votes
0answers
116 views

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. ...
7
votes
0answers
401 views

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 ...
6
votes
0answers
91 views

Justify an unbiased estimator is UMVUE

Suppose $X_1,\ldots,X_n$ are iid $N(\theta,\theta)$, with $\theta\in(0,\infty)$. Is $\bar{X}$ the UMVUE (beta unbiased estimator) of $\theta$? I find the complete sufficient statistic is ...
6
votes
0answers
136 views

Normalizing factor for product of Gaussian densities - interpretation with Bayes theorem

The normalizing factor for the product of two multivariate Gaussian densities, $f(x)$ and $g(x)$ with mean vectors $a$ and $b$ respectively, and covariance matrices $A$ and $B$ respectively, is itself ...
5
votes
0answers
3k views

Uniform distribution on the surface of unit sphere

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere. Intuitively, I know that that's ...
5
votes
0answers
89 views

Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?

I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants): $$ \log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
4
votes
0answers
187 views

The distribution of the inner product of a random complex normal vector.

Good day! I would like to find the distribution of the inner product of a random complex normal vector with: some constant vector; random gaussian vector. Let's assume a vector $\vec{z}$ which has ...
4
votes
0answers
190 views

Characterization of the law of a stochastic process by its finite dimensional distributions

Let $(\Omega,\mathcal{A},\mathbb{P})$ a probability space. Let $(X_t)_{t \in [0,T]}$, $(Y_t)_{t \in [0,T]}$ (real-valued) centered Gaussian processes such that the finite dimensional distributions ...
4
votes
0answers
77 views

Teaching Student's distribution

While it is fairly straightforward to show the basics of the normal distribution in a first year undergraduate course, how does a teacher provide good intuition when the Student distribution comes in? ...
3
votes
0answers
53 views

Compound Distribution — Log Normal Distribution with Log 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 (actually, the mean ...
3
votes
0answers
31 views

Compound Distribution — 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 Normal Distribution whose mean is distributed Normally. ...
3
votes
0answers
107 views
3
votes
0answers
198 views

Joint pdf of N > 1 i.i.d. random variables isotropic if and only if they are centered gaussian?

Are centered Gaussian densities given by $$f_X(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-x^2/(2 \sigma^2)}$$ the unique densities such that the joint pdf of $N > 1$ independent and identically ...
3
votes
0answers
261 views

Covariance matrix and Gaussian i.i.d. random variables

I have a set $X = \left \{ X_i | i \in (1,n) \wedge X_i \text{ is a random variable} \right \} $ Does $\forall i \in (1,n ), X_i \text{ follows a normal distribution} $ implies that ...
3
votes
0answers
68 views

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 ...
3
votes
0answers
35 views

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 $F_{Y_n}$ ...
3
votes
0answers
157 views

Most powerful test for discrete variable

The discrete random variable X has the following probability distributions under $H_0$ and $H_1$ $$\begin{array}{r|rrrrrrrrrr} x&1&2&3&4&5&6&7&8&9&10\\ ...
3
votes
0answers
863 views

Integral of a random process that follows Gaussian Process

Suppose $X(t)$ follows a strictly-sense stationary(SSS) Gaussian Process with the mean to be $\mu$ and autovariance $\sigma^2$ How to prove that $\int_{0}^{T}{{X(t)}dt}$ is random variable that ...
3
votes
0answers
140 views

How to integrate the following formula about normal distribution

How to compute the following formula? $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, dx $$ $$ \int_{-\infty}^{+\infty} \Phi(x) N(x\mid\mu,\sigma^2) \, xdx $$ where ...
3
votes
0answers
132 views

Exponentials of chi-squared random variables (and their sums)

Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. chi-squared random variables with $t$ degrees of freedom, i.e. $X_i\sim\chi^2_t$. I am wondering what is known about the distribution of ...
3
votes
0answers
82 views

Is there an algebraically normal function from $\mathbb{Z}^{2}$ to $\{ 0 , 1\}$?

Let $\gamma : \mathbb{R} \to \mathbb{R}^{2}$ be a real algebraic curve. Let $r \geq 0$ and $I \subset \mathbb{R} $ then $\gamma_{r} (I)= \{a \in \mathbb{R}^{2} : \exists b \in \gamma(I), d(a,b)\leq r ...
3
votes
0answers
117 views

Integral of the Normal Characteristic Function

The characteristic function of the $N$-variate Normal distribution is $$\forall \mathbf{t} \in \mathbb{R}^N \quad \psi(\mathbf{t}) \equiv \mathbb{E}\left( e^{i\mathbf{t}X}\right) = \exp \left( i{ ...
3
votes
0answers
2k views

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 ...
3
votes
0answers
169 views

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 ...
3
votes
0answers
70 views

Unknown result in probability theory relating CDF of any density to the CDF of normal distribution

There is apparently a result in probability theory saying: If $A(z)$ is any cumulative distribution function, $\alpha(t)$, the corresponding characteristic function and $\Phi(z) = ...
2
votes
0answers
46 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal ...
2
votes
0answers
33 views

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 ...
2
votes
0answers
46 views

Probabilty concerning two normal distributed random variables

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 ...
2
votes
0answers
47 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 ...
2
votes
0answers
47 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 ...
2
votes
0answers
41 views

Ammunition Depot: Monte Carlo Method

I was given the following question from a friend of mine and I can't seem to understand it to well: A squadron of 10 bombers attempts to destroy an ammunition depot. The fighter jet flies in the ...
2
votes
0answers
122 views

Nearest neighbor for planar Poisson is normally distributed

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 ...
2
votes
0answers
30 views

Proof of Conjugacy Between Multiple Multivariable Normal Distributions and Normal Inverse Wishart Distribution

I've been trying to prove that a normal inverse Wishart distribution can act as a conjugate to a series of multivariable normal distributions. Formally, $$\prod_{i=1}^I Norm_{\boldsymbol ...
2
votes
0answers
56 views

Help solving integration: $I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(a/\sqrt{b+c\mathrm{e}^{\frac{x-\mu}{\sigma}}}\right)dx$

My work has arrived at needing to solve the integral below for $a,b,c,\sigma>0$ $$I=\int_{-\infty}^{\infty}\phi\left(x\right)\Phi\left(\frac{a}{\sqrt{b+c\mathrm{e}^{(x-\mu)/\sigma}}}\right)dx$$ I ...
2
votes
0answers
54 views

Help Integrating $I=\int\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$

I am trying to integrate the following function involving the Normal CDF ($\Phi$). I actually need the definite integral $$\int^b_a\Phi\left(\frac{p}{\sqrt{q+rx}}\right)dx$$ for $q+ra,q+rb >0$ but ...
2
votes
0answers
41 views

One-sided Bound on Sum of Fourth Moments

I'm interested in methods for proving one-sided bounds of the form $$ \mathbb{P}[\frac{1}{n}\sum_{i=1}^n X^4_i \geq 3+t]\leq Ce^{-nt} $$ where $X_i$ are standard normal random variables. I've run a ...
2
votes
0answers
46 views

Uniform convergence of functions involving normal CDF

Consider two sequences of continuous functions $(f_n)$ and $(g_n)$ for $n \geq 0$ defined by $$ f_n (x) := \int_0 ^t \Phi\left(\frac{x\Phi ^{-1}(\alpha(s) + \beta_n(s))+\Phi^{-1} ...
2
votes
0answers
51 views

what is the expectation of $\sqrt{\left | x \right |} * sign(x)$ and $log(|x|)$ for a normal distribution

(1) What would $\int_{-\infty }^{\infty} \frac{\sqrt{\left | x \right |} * sign(x)}{\sqrt{2\pi}\sigma}e^{-0.5*\left ( \frac{x-\mu}{\sigma} \right )^{2}}dx$ evaluate to? This is expectation of ...
2
votes
0answers
53 views

Prove or disprove that the Bhattacharyya distance is a true distance function

Let $\mathcal{X}\equiv\Bbb{R}^n\times\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ is the space of all symmetric positive-definite $n\times n$ real matrices. Let $x,y\in\mathcal{X}$, where $$ ...
2
votes
0answers
34 views

Distribution of $\langle A,x\rangle\langle A,y\rangle + \langle B,x\rangle\langle B,y\rangle$ given $\langle x, y\rangle$

Let $A$ and $B$ be independent, normal distributed $N(0,1)$ normalized unit vectors, and let $x$ and $y$ be unit vectors with given inner product $\langle x, y\rangle=u$. Can we write the ...
2
votes
0answers
41 views

Solving a nonlinear constrained optimization involving CDF and expectation of normal distribution

I would like to know if it is possible to solve the following nonlinear constrained optimization problem and find how the optimal solution varies with $C$ and $\beta$: $\max_{x,y}\beta ...
2
votes
0answers
46 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...
2
votes
0answers
27 views

Distribution of sample statistics taken from bivariate normal

$(X_{1},Y_{1}),\,...\,,(X_{n},Y_{n})' (n>2)$ are random samples taken from $N_{2}((\mu_{1},\,\mu_{2})',\,$$ \begin{pmatrix} \sigma^{2}_{1} & \rho\sigma_{1}\sigma_{2} \\ ...
2
votes
0answers
30 views

Distribution of some linear combination of Normal RVs

I would like to ask for help concerning this question lifted from the book An Introduction to the Theory of Statistics by Mood, Graybill, and Boes (2nd ed.). Let $X_1$ and $X_2$ be independent ...
2
votes
0answers
69 views

Could we define two random variables such that the product of them is Normal distribution(Gaussian)?

Could we find two random variables $X$ and $Y$ which $XY \sim N(\mu, \sigma^2)$? I found the ratio of two normal distributed random variables is distributed Cauchy distribution. However, on the ...
2
votes
0answers
32 views

Characteristics function and moments of multivariates

I have been reading this paper recently-- http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.199.2157&rep=rep1&type=pdf This is a paper by Nengjiu Ju who uses talyor series to express ...
2
votes
0answers
90 views

Normal pdf/cdf inequality

Let $\Phi$ be the cdf and $\phi$ the pdf of the standard normal distribution. I want to show that: $$ \Phi(z)[1-\Phi(z)]\geq \phi(z)^2, \quad z\in\mathbb R. $$ How can I do this? I tried by looking at ...
2
votes
0answers
41 views

How to calculate this kind of probability for a normal distribution?

here is my question. I have a normal distribution with known mean and variance. Say the mean is 3 and the Var. is 2. what is the probability that the random variable is taking value 2.9? If I plug ...
2
votes
0answers
125 views

Finding the limiting probability distribution

I found this problem in Shiryaev's Problems in probability (Problem 3.4.14). Let $\xi_1, \xi_2, \dots$ be a sequence of independent and $N(0, 1)$-distributed random variables. Setting $S_n = ...
2
votes
0answers
36 views

Best line fit for correlated points

Given in $\mathbb{R}^3$ are $n$ points $\mathbf{y}_i\sim N(\mathbf{y}_i-\mathbf{\hat{y}}_i, \mathbf{C}_i)$, which are normally distributed. I want to determine a best fit line $\mathbf{u}(\lambda) = ...