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

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41
votes
9answers
6k views

What do $\pi$ and $e$ stand for in the normal distribution formula?

I'm a very beginner in mathematics and there is one thing I've been wondering recently. The formula for the normal distribution is: ...
37
votes
2answers
13k views

Expectation of the maximum of gaussian random variables

Is there an exact or good approximate expression for the expectation, variance or other moments of the maximum of $n$ independent, identically distributed gaussian random variables where $n$ is large? ...
32
votes
2answers
5k views

Why is the error function defined as it is?

$\newcommand{\erf}{\operatorname{erf}}$ This may be a very naïve question, but here goes. The error function $\erf$ is defined by $$\erf(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2}dt.$$ Of ...
21
votes
4answers
36k views

Is the product of two Gaussian random variables also a Gaussian?

Say I have $X \sim \mathcal N(a, b)$ and $Y\sim \mathcal N(c, d)$. Is $XY$ also normally distributed? Is the answer any different if we know that $X$ and $Y$ are independent?
15
votes
1answer
2k views

How was the normal distribution derived?

Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as ...
15
votes
2answers
471 views

How to generate points uniformly distributed on the surface of an ellipsoid?

I am trying to find a way to generate random points uniformly distributed on the surface of an ellipsoid. If it was a sphere there is a neat way of doing it: Generate three $N(0,1)$ variables ...
14
votes
3answers
265 views

If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$

Let $A$ be a positive definite real $n\times n$ matrix. How can I prove that $$ \int_{\mathbb{R}^n}\mathrm{e}^{-\langle ...
11
votes
1answer
1k views

Cool examples of the Central Limit Theorem in action

Sir Francis Galton has described the Central Limit Theorem as follows. I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the "Law of ...
10
votes
3answers
23k views

How to calculate the integral in normal distribution?

The factory is making products with this normal distribution: $\mathcal{N}(0, 25)$. What should be the maximum error accepted with the probability of 0.90? [Result is 8.225 millimetre] How will I ...
10
votes
2answers
990 views

Why don't we allow the canonical Gaussian distribution in infinite dimensional Hilbert space?

I'm looking at Gaussian distributions in infinite-dimensional Hilbert space, and the sources I've seen so far say that the covariance matrix has to be of trace class (i.e. the trace must be finite). ...
10
votes
0answers
103 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. ...
9
votes
3answers
753 views

Is the mean of the truncated normal distribution monotone in $\mu$?

I am wondering whether the mean of the truncated normal distribution is always increasing in $\mu$. The untruncated distribution of $x$ is $\mathcal{N}(\mu,\sigma^2)$. The mean of the truncated ...
9
votes
1answer
288 views

Solution of differential equation related to Normal density

Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ Given $0<\sigma\le 1$. I wish to know whether there ...
9
votes
1answer
138 views

Characterization of normal distribution

I am sorry if this question is vague since I am completely unfamiliar with probability theory. Suppose that we have a family of real-valued random variables $X_n$ (say, all of them have mean 0) on ...
9
votes
1answer
479 views

length of Gaussian Random Vector

Suppose I have a random vector $x=[x_1,...,x_k]$ s.t. $x∼N(\mu,\sum)$. How is the length or magnitude of $x$ distributed? I know that if $k=2$ and $\sigma_1=\sigma_2$ and $\sigma_{12}=0$ ($x_1$ and ...
9
votes
1answer
123 views

Inequality for $N(0,1)$ CDF: $|\log F(v)|\leq |\log F(0)|+|v|+|v|^2$

Suppose that $F$ is the CDF of a standard normal distribution. Hayashi (2000) claims that the following is true $$ |\log F(v)|\leq |\log F(0)|+|v|+|v|^2\quad\text{for all}\quad v. $$ How does ...
9
votes
1answer
591 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
8
votes
1answer
14k views

Calculation of the n-th central moment of the normal distribution $\mathcal{N}(\mu,\sigma^2)$

Since integration is not my strong suit I need some feedback on this, please: Let $Y$ be $\mathcal{N}(\mu,\sigma^2)$, the normal distrubution with parameters $\mu$ and $\sigma^2$. I know $\mu$ is the ...
8
votes
2answers
8k views

Product of Two Multivariate Gaussians Distributions

Given two multi-variate gaussians distrubtions, given by mean & covariance, G1(m1,sigma1) & G2(m2,sigma2), what are the formulae to find the product i.e G1 * G2 ? And if one was looking to ...
8
votes
2answers
219 views

Joint distribution of the signs of the partial sums of independent standard normal random variables

Consider some i.i.d. standard normal random variables. What is the joint distribution of the signs of their partial sums? More formally, define a sequence of random variables ...
8
votes
2answers
323 views

Computing the Gaussian integral with step functions

Say, we are interested in deriving $$\int_{-\infty}^{\infty}e^{-x^2}=\sqrt{\pi}\tag{1}$$ There are many well known ways to do it, for example: by polar coordinates via the gamma function, etc. ...
8
votes
2answers
534 views

Why should Gaussian noise have fractal dimension of 1.5?

In a paper I'm trying to understand, the following time series is generated as "simulated data": $$Y(i)=\sum_{j=1}^{1000+i}Z(j) \:\:\: ; \:\:\: (i=1,2,...,N)$$ where $Z(j)$ is a Gaussian noise with ...
7
votes
1answer
7k views

Probability of a point taken from a certain normal distribution will be greater than a point taken from another?

Let's say I have one point that will be taken randomly from a normal distribution with mean $\mu_1$ and standard deviation $\sigma_1$. Let's say I have another point that is taken much in the same ...
7
votes
4answers
4k views

Derivation of the density function of student t-distribution from this big integral.

My lecturer posed a question where we derive the density function of the student t-distribution from the Chi-square and Standard normal distribution. I worked on this question for days, and I am ...
7
votes
1answer
202 views

Concentration of measure bounds for multivariate Gaussian distributions (fixed)

Let $\gamma_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. It is known (see for example Cor 2.3 here: http://www.math.lsa.umich.edu/~barvinok/total710.pdf) that ...
7
votes
0answers
342 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
3answers
239 views

How do I evaluate $\int \limits_{-\infty}^{a} e^{−t^2}dt$?

I know that $$I \equiv \int \limits_{-\infty}^\infty e^{−t^2} \, dt=\sqrt{\pi},\text{ and }\int \limits_{-\infty}^0 e^{−t^2} \, dt=\frac{\sqrt{\pi}}{2}.$$ However, I don't understand if (or how) I ...
6
votes
4answers
109 views

If $X \sim N(0,1)$, why is $E(X^2)=1$?

If $X$ is a normally distributed with mean $0$ and variance $1$, expectation of $X$ equals $0$ but why is $E(X^2)=1$?
6
votes
1answer
3k views

Linear transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of ...
6
votes
1answer
188 views

An interesting inequality about the cdf of the normal distribution

When approaching this other question I came out with the inequality: $$\frac{1}{4+x^2}e^{-x^2/2} \leq\Phi(x)\Phi(-x)\leq \frac{1}{4}e^{-x^2/2},\tag{1}$$ where $\Phi(x)$ is the cdf of the standard ...
6
votes
1answer
956 views

Can the product of two non-independent Gaussians be Gaussian?

We recently discussed this: Is the product of two Gaussian random variables also a Gaussian? What was established was that in nontrivial cases (i.e., ruling out zero-variance Gaussians, which are ...
6
votes
1answer
82 views

Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
6
votes
1answer
106 views

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
6
votes
1answer
292 views

Fractional Part of Sum of Sequence of Independent Normal Random Variables

I'm trying to prove that if $X_n$ iid normal $S_n = \sum_1^n X_i$ $U_n=S_n-\lfloor S_n\rfloor$ then $U_n$ is asymptotically uniform in distribution. I've got no idea how to approach this, and it's ...
6
votes
0answers
61 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
108 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
2answers
288 views

Triangular vs Normal distribution

I'm trying to approximate a standard normal distribution with a triangular distribution. What parameters of the triangular distribution (min, max and mode) are more suitable? Thank you
5
votes
4answers
690 views

Sampling from a $2$d normal with a given covariance matrix

How would one sample from the $2$-dimensional normal distribution with mean $0$ and covariance matrix $$\begin{bmatrix} a & b\\b & c \end{bmatrix}$$ given the ability to sample from the ...
5
votes
1answer
13k views

What is the expectation of $ X^2$ where $ X$ is distributed normally?

I know that if $X$ were distributed as a standard normal, then $X^2$ would be distributed as chi-squared, and hence have expectation $1$, but I'm not sure about for a general normal. Thanks
5
votes
4answers
7k views

Asymmetric Normal Probability Distribution

I'm looking for a continuous probability distribution a little bit like the normal distribution but asymmetric. In my opinion this distribution applies to phenomenons related to response time in ...
5
votes
1answer
8k views

'normally distributed random numbers' vs 'uniformly distributed random number'?

what is the difference between 'normally distributed random numbers' and 'uniformly distributed random number'? A answer in a layman language is appreciated :)
5
votes
3answers
169 views

How to Make a PDF 'Look' Uniform?

Let $X$ be a normally-distributed random variable with mean zero and variance $\sigma^2$: $X \sim N(0,\sigma^2)$. Let $Y$ be a mapping from $X$ onto the interval $(0,1)$ using the sigmoid function: ...
5
votes
1answer
3k views

Multivariate Normal Difference Distribution

Since the distribution of a difference of two normally distributed variates X and Y with means and variances $(\mu_x,\sigma_x^2)$ and $(\mu_y,\sigma_y^2)$ respectively is given by another normal ...
5
votes
1answer
166 views

Estimating a gaussian distribution from a GMM

Suppose that we have a Gaussian mixture model (GMM) in n-dimensional space: $$P_1(x) = \sum_{i=1}^{C}\pi(c_i)\mathcal{N}(\mu_i,\Sigma_i)$$ We want to estimate a single Gaussian distribution from ...
5
votes
2answers
190 views

expectation equations

I am just trying to understand the following three equations. $\phi(x)$ denotes the standard Gaussian cumulative distribution function and $X$~$N(\mu,\sigma^2)$ (1) $\mathbb{E}[e^{tX}f(X)]=e^{\mu ...
5
votes
2answers
135 views

Convolution with Gaussian, without distribution theory, part 1

I only know basic $L^p$ theory (nothing about distributions) and am trying to prove the following: Let $t>0$, $f\in L^{p}(\mathbb{R}^n,m)$, $\Gamma(t,x)=(4\pi t)^{-n/2}e^{-|x|^2/4t}$ and $$ ...
5
votes
1answer
471 views

$\int_0^tB_s^2\ dB_s$ - Gaussian Process and independent increments?

For $(B_t)_{t\ge0}$ a standard Brownian motion (Wiener process) define the stochastic process $X_t:=\int_0^tB_s^2\ dB_s$. I am currently trying to assess if $(X_t)_{t\ge0}$ is a Gaussian process and ...
5
votes
2answers
517 views

Convergence of a sequence involving the maximum of i.i.d. Gaussian random variables

It's well known that, for a sequence of $n$ i.i.d. standard Gaussian random variables $X_1,\ldots,X_n$, where $X_\max=\max(X_1,\ldots,X_n)$, the following convergence result holds: ...
5
votes
3answers
350 views

Central Limit Theorem Definition

My friend and I have a bet going about the definition of the Central Limit Theorem. If we define an example as a number drawn at random from some probability density function where the function has a ...
5
votes
2answers
4k views

Distribution of the maximum of a multivariate normal random variable

Suppose there is a vector of jointly normally distributed random variables $X \sim \mathcal{N}(\mu_X, \Sigma_X)$. What is the distribution of the maximum among them? In other words, I am interested in ...