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

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38
votes
9answers
4k 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: ...
29
votes
2answers
4k 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 ...
22
votes
2answers
8k 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? ...
12
votes
3answers
19k 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?
10
votes
1answer
705 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 ...
9
votes
3answers
470 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 ...
8
votes
2answers
740 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). ...
8
votes
1answer
201 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 ...
8
votes
2answers
226 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
326 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 ...
8
votes
1answer
281 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 ...
7
votes
3answers
2k 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 ...
6
votes
4answers
102 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
3answers
222 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
1answer
7k 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 ...
6
votes
1answer
427 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 ...
6
votes
1answer
4k 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 ...
6
votes
1answer
740 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
0answers
229 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 ...
5
votes
3answers
13k 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 ...
5
votes
4answers
355 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
4answers
4k 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
3answers
153 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
97 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
176 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
1answer
305 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
260 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
1answer
3k views

X,Y are independent standard normal distributed then what is the distribution of $\frac{X}{X+Y}$

X, Y are independent standard normal random variables, what is the distribution of $$ \frac{X}{X+Y} $$ Could anyone help me with this? Thanks. I have worked the problem by multivariable ...
5
votes
3answers
275 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
1answer
177 views

Expectations containing normal CDF

Suppose that $X\sim\mathcal{N}\left(0,1\right)$ (i.e., $X$ is a standard normal random variable) and $a,b,$ and $c$ are some real constant. Does any of the following expectations have a closed-form? ...
5
votes
1answer
67 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 ...
5
votes
3answers
555 views

Compute probability of a particular ordering of normal random variables

There are $m$ normally distributed, independent random variables $N_1, \ldots, N_m$ with distinct means $\mu_1, \ldots \mu_m$ and standard deviations $\sigma_1, \ldots, \sigma_m$. Then, we get a ...
5
votes
1answer
128 views

how to evaluate a definite integral (looks almost like nonintegral moments of a Gaussian)

I'd like to show the following equality (at least Mathematica claims it is an equality): \begin{multline*} \int_0^\infty x^p \exp(-(ax - b)^2)\, dx = \frac{1}{2} e^{-b^2} a^{-p-1} \left(\Gamma ...
5
votes
1answer
213 views

A case of the central limit theorem

I want to show that $$\frac{\sum_{k=1}^N X_k}{\sqrt{\sum_{k=1}^N X_k^2}} \overset{N\to\infty}{\to} \mathcal{N}(0,1)\text{ in distribution,}$$ where $X_1,X_2,\ldots$ is a sequence of iid random ...
5
votes
1answer
56 views

Why does adding 3 random decimals in the range [-1,1] give a normal dist with std. dev 1?

I've used Math.random()*2-1+Math.random()*2-1+Math.random()*2-1 many times in the past to get normally-distributed random numbers with a standard deviation of 1. ...
5
votes
1answer
205 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 ...
5
votes
0answers
75 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 ...
5
votes
1answer
293 views

Does this table fit the normal distribution?

The Pascal triangle can be described by the recurrence: $P(n,1)=1, k>1: P(n,k) = P(n-i,k-1) + P(n-i,k)$ This well known triangle has the basic properties that the ratios of consecutive ...
4
votes
2answers
216 views

A simpler solution of the integral $\int_{x_1+\ldots+x_n \geq a} \exp\left[ -\pi \left(x_1^2+\ldots+x_n^2 \right)\right] dx_1\cdots dx_n $

I want to calculate the following integral $$\int_{x_1+\ldots+x_n \geq a} \exp\left[ -\pi \left(x_1^2+\ldots+x_n^2 \right)\right] dx_1\cdots dx_n, $$ as a function of $a$, in possibly the shortest ...
4
votes
2answers
734 views

Connection to Normal distribution

I've been working on finding the probability for the event, that the sum of $n$ independent random variables are less than $s$, when they are evenly distributed on $[0,1)$. I've used the law of total ...
4
votes
2answers
275 views

Let $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. Find $EZ$.

Let $X,Y$ independent random variables with $X,Y\sim \mathcal{N}(0,1)$. Let $Z=\max(X,Y)$. I already showed that $F_Z$ of $Z$ suffices $F_Z(z)=F(z)^2$. Now I need to find $EZ$. Should I start like ...
4
votes
4answers
5k views

Calculate the expected value of $Y=e^X$ where $X \sim N(\mu, \sigma^2)$

I got a problem of calculating $E[e^X]$, where X follows a normal distribution $N(\mu, \sigma^2)$ of mean $\mu$ and standard deviation $\sigma$. I still got no clue how to solve it. Assume $Y=e^X$. ...
4
votes
1answer
2k 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 ...
4
votes
1answer
186 views

Expectation value of $1/x$

Given a random variable $x$ which is assumed to follow a Gaussian distribution $x \sim N( \mu, \sigma^2 )$ and $x$ is further known to be positive, I am interested in the following expectation value: ...
4
votes
1answer
385 views

Generate a set of random numbers with a normal distribution

I am trying to generate a set of N random numbers where the set has a normal distribution. I'm currently using a brute force approach: Randomly select N numbers from a normal distribution. Check ...
4
votes
1answer
140 views

Probability of a gaussian distribution in another gaussian distribution

Assume we have a Gaussian distribution $p(x) \sim \mathcal{N}(\mu_p,\Sigma_p)$ For any point $X$, it is easy to compute the density of $x$ in $p$: $$p(x) = \frac{1}{|2\pi ...
4
votes
1answer
93 views

Tricky probability problem

I am having trouble with proving the following assertion: $X,Y$ are i.i.d. with mean $0$ and variance $1$. If $X+Y$ and $X-Y$ are independent then $X,Y$ are normally distributed. Should I be ...
4
votes
2answers
108 views

Fraction Problem. 3rd grader question got parents thinking

So our nine year old son comes home from 3rd grade and tells us an amazing thing happened in school today. He was playing a math game with his friend and they got the same score two times in a row! ...
4
votes
2answers
322 views

Distribution of $Y = \sin X$ when $X$ is normal?

Assume $X$ is Normally distributed : $X\sim N(\mu,\sigma)$ What is the distribution of $Y = \sin X$ ? I think we should start with $F_Y(y)=P(\sin X < y)$. But how to continue?
4
votes
2answers
101 views

Central Limit Theorem. How to apply to the task.

The research showed that the probabilities of 3, 4, 5, 6 and 7 cars broken on one day are 0.3, 0.4, 0.2, 0.08, 0.02 respectively. If 221 car broke in 50 days, does it show that more cars break than ...