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

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1
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1answer
173 views

How to apply Gaussian kernel to smooth density of points on 2D (algorithmically)

I have a set of points on a 2D surface and need to build a heatmap. However, I also need to smooth out the density/distribution by applying some sort of kernel (Gaussian kernel, for example). I Know ...
1
vote
2answers
714 views

Sum of two truncated gaussian

What is the CDF and the PDF (or approximation) of the sum of two truncated gaussian $X = TN_x(\mu_x,\sigma_x;a_x,b_x)$ and $Y = TN_y(\mu_y,\sigma_y;a_y,b_y)$ ? where $TN(\mu,\sigma;a,b)$ is a ...
1
vote
3answers
35 views

Show that $Y\sim N(a+A\mu,AVA^T)$

Consider $Z=(Z_1,\ldots,Z_n)^T\sim N(\mu,V)$. Show: If $a\in\mathbb{R}^m$ and $A$ is a $(m\times n)$-matrix with $\text{rang}(A)=m$ then $$ Y=a+AZ\sim N(a+A\mu,AVA^T). $$ My ...
1
vote
1answer
523 views

Mean and Variance Convergence with r.v.

Let $(X_n)_{n\ge 1}$ be a sequence of random variables, with respective distributions being Gaussian, with respective mean $\mu_n \in \mathbb R$ and variance $\sigma_n^2 > 0$. Prove that if $X_n$ ...
1
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2answers
15k views

Calculating mu and sigma (μ and σ) of a normal random variable

Let X be a normally distributed variable with unknown parameters μ and σ (sigma). If we know that P (X ≥ 75) = 0.7291 and P (X ≥ 83) = 0.7764. With the information given Is it possible to determine ...
1
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1answer
533 views

How to directly compute an integral which corresponds to the normal distribution

How does one directly (by finding primitive) compute an integral which corresponds to the normal distribution: $$\int_{a}^{b} e^{{-(x-a)^2}/{2s^2}} \,\mathrm{d}x$$
0
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1answer
67 views

Product of two densities, when one of them is “incomplete”

One can frequently read that the product of two multivariate Gaussian pdfs, $f_1(x)$*$f_2(x)$, is itself a Gaussian function, with parameters as defined for example in: ...
0
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1answer
293 views

Linear transformation of random variables

We have to stochastic variables X and Y, and we define $ \begin{pmatrix} \tilde{X} \\ \tilde{Y} \end{pmatrix}=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \begin{pmatrix} X \\ Y ...
0
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1answer
61 views

Normal distribution tail probability inequality

I am trying to show that $$P(X>t)\leq \frac{1}{2}e^\frac{-t^2}{2}$$ for $t>0$ where $X$ is a standard normal random variable. Perhaps this is simple. I have been starting with $$ ...
0
votes
2answers
500 views

Proof that if $Z$ is standard normal, then Z^2 is distributed Chi-Square (1).

Suppose that $Z\sim N(0,1)$ and let $V=Z^2$. Prove that $V\sim \chi^2(1)$. I want to use the method of moment generating functions, because I already understand the proof using the method of ...
0
votes
1answer
104 views

Transforming distributions

There is an economy, populated by a large number of agents. A first order condition common to all agents, is the following: $$E[\exp^{(1-\theta)\eta_i}(r-R+\eta_i)]=0$$ the index $i$ indicates the ...
0
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1answer
54 views

Probability question of independent random varaibles

Let $X\sim \mathcal{N}(6,1)$ and $Y\sim\mathcal{N}(7,1)$ be two independent normal variables. Find $Pr(X>Y)$. the answer is $0.2389$ but I do not know how to do it.
0
votes
1answer
68 views

Confusion related to gaussian distribution

I was reading this paper where it had a gaussian distribution model. I mean gaussian is given by $P(y) = \frac{e^{-\frac{1}{2}(y -\mu)^T \Sigma^{-1}(y -\mu)}}{2\pi^{n/2}|\Sigma|^{1/2}}$ But is ...
0
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1answer
892 views

equivalence between uniform and normal distribution

The principle of insufficient reason says that all outcomes are equiprobable when we have no knowledge to guess otherwise. I understand this and that this corresponds to uniform distribution. However, ...
23
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2answers
4k 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 ...
10
votes
2answers
11k 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 ...
11
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2answers
1k 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). ...
5
votes
1answer
615 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 ...
4
votes
1answer
330 views

Integral of a Gaussian process

Let $(\Omega,\Sigma,P)$ be a probability space and $X: [0,\infty) \times \Omega \to \mathbb{R}$ be a Gaussian process (i.e. all finite linear combinations $\sum_i a_i X_{t_i}$ are Gaussian random ...
4
votes
1answer
144 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 ...
7
votes
1answer
19k 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
3
votes
3answers
4k views

$X$ standard normal distribution, $E[X^k]=?$

I'm stuck with a homework problem where we are supposed to prove that the expected value $E[X^k]$, if $X$ has standard normal distribution, is equal to: $$E[X^{2k}]=\frac{(2k)!}{k!\cdot2^k}.$$ But I ...
2
votes
1answer
80 views

Compound Distribution — Normal Distribution with Log Normally Distributed Variance

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 variance is distributed Log ...
17
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2answers
1k 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 ...
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 ...
3
votes
0answers
67 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. ...
9
votes
1answer
702 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 ...
6
votes
1answer
238 views

Berry-Esseen bound for binomial distribution

From the Berry-Essen theorem I can deduce $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right| \le \frac{C(p^2+q^2)}{\sqrt{npq}}$$ with $C \le 0.4748$. My ...
6
votes
2answers
302 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 ...
5
votes
1answer
254 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? ...
4
votes
2answers
2k views

Convolute exponential with a gaussian

I have data measuring an exponential decay that is convoluted by a gaussian response function. I have the measured shape of the gaussian, and want an analytical expression for the exponential ...
2
votes
0answers
47 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
1answer
162 views

Distribution of largest sample from normal distribution.

Given $n$ independent random variables $X_i$ with normal distribution, mean $\mu$, variance $\sigma^2$, what is the distribution of $\max\limits_{i=1}^n(X_i)$ ? In particular I am interested in ...
2
votes
2answers
1k views

Determining distribution of maximum of dependent normal variables

I have a stochastic variable x with this property: if it's measured at t1 and again at t2, then x(t2)-x(t1) has a normal distribution with mean 0 and standard deviation Sqrt[t2-t1]. I want to find ...
1
vote
1answer
66 views

Compound Distribution — Uniform Distribution with Normally Distributed Parameters

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 Uniform Distribution whose parameters are distributed ...
7
votes
1answer
289 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
4answers
18k views

Combining two probability distributions

I have a variable $X$. In a measurement $A$, $X$ follows the normal distribution $N_1$ with mean $m_1$ and standard deviation $\sigma_1$. In a similar measurement $B$, $X$ follows another normal ...
5
votes
2answers
685 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
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
1answer
5k 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 ...
4
votes
2answers
6k views

Normal Distribution, The “Y” Value

Guys I am having trouble with the standard normal distribution. http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalLesson.htm We know the X values run from approx $-\infty$ to $+\infty$ but ...
4
votes
1answer
437 views

Does the integral of PDF of multi-normal distribution over quarter planes have a closed form?

I am interested in finding a closed form solution (wich I suspect does not exist) to the following integral $$\displaystyle \int _a^{\infty }\int _b^{\infty } \frac{\exp \left(-\frac{x^2+y^2-2 c x ...
4
votes
1answer
7k views

Expected value of normal distribution given that distribution is positive

Given $X \sim N(0, \sigma^2)$ (that is, $X:\mathbb{R} \to \mathbb{R}$ is a normal random variable with mean $0$ and variance $\sigma^2$), I'm trying to calculate the expected value of $X$ given that ...
4
votes
2answers
460 views

Nested normal-distribution integral

Is there an analytical or approximate solution of the following integral? $$ \int_{-\infty}^{\infty}\int_{y-d}^{y+d}\exp\big(-{(x-\mu_1)^2}/{2\sigma^2}\big) \exp\big(-{(y-\mu_2)^2}/{2\sigma^2}\big) ...
3
votes
1answer
70 views

concentration of maximum of gaussians

Let $X=(X_1,\ldots,X_n)$, where $X_i \sim N(0,1)$ are iid. I'm looking for a result (and a proof outline) on the concentration of the max abs value of these Gaussians, $\|X\|_\infty$. That is, some ...
3
votes
1answer
186 views

Normal approximation of tail probability in binomial distribution

From the Berry Esseen theorem I know, that $$\sup_{x\in\mathbb R}|P(B_n \le x)-\Phi(x)|\in O\left(\frac 1{\sqrt n}\right)$$ whereby $B_n$ has the standardized binomial distribution and $N$ has the ...
3
votes
1answer
303 views

Bound on the $Q$ function related to Chernoff bound

For the function $Q(x) := \mathbb{P}(Z>x)$ where $Z \sim \mathcal{N}(0,1)$ \begin{align} Q(x) = \int_{x}^\infty \frac{1}{\sqrt{2\pi}} \exp \left(-\frac{u^2}{2} \right) \text{d}u, \end{align} for ...
3
votes
1answer
57 views

Truncated Mean Squared

Suppose that $X_{\sigma} \sim \mathcal{N}(\mu,\sigma^{2})$. I am interested in whether $f(\sigma)=\mathbb{E} (X_{\sigma}^2 1_{\{X_{\sigma}>0\}})$ is monotonic in $\sigma$ for all $\mu$. I ...
2
votes
0answers
24 views

Joint Density and Covariance between Two Random Variables with the same Mean and Variance

This seems like a deceptively simple question, (and it perhaps is and I am missing something) but I could not find anything on this. Q1) Are there any general results / relationships to get the ...
2
votes
0answers
65 views

Simple question on conditional probabilities (multidimensional normal distributions)

Let $X$ and $Y$ in $\Bbb{R}^n$ be two random vectors. We assume that $X\mid Y\sim\mathcal{N}(Y,\Sigma_X)$ and $Y\sim\mathcal{N}(\mu_Y,\Sigma_Y)$ The goal is to sample from the distribution of $X$. ...