For questions about the Gaussian probability distribution, its definition, properties and use.

# Questions tagged [gaussian]

555 questions

**37**

votes

**7**answers

### Looking for a function that approximates a parabola

I have a shape that is defined by a parabola in a certain range, and a horizontal line outside of that range (see red in figure).
I am looking for a single differentiable, without absolute values, non-piecewise, and continuous function that can…

Gimelist

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**10**

votes

**1**answer

### Evaluating the integral $ \int_0^1 \frac{e^{-y^2(1+v^2)}}{(1+v^2)^n}dv$

I am trying to evaluate the integral
$$
\int_0^1 \frac{e^{-y^2(1+v^2)}}{(1+v^2)^n}dv = e^{-y^2}\int_0^1 \frac{e^{-y^2v^2}}{(1+v^2)^n}dv
$$
for $n\in \mathbb{N}$.For n=1 one finds Owen's T function, i.e.
\begin{align}
\int_0^1…

drandran12

- 462
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**7**

votes

**0**answers

### An inequality involving two i.i.d. standard Gaussians.

Short question:
Let $ r,t,h\ge0 $ and $ G_1,G_2 $ be i.i.d. standard Gaussians.
Is it true that $$ \Pr[\{(r+G_1)^2\le t\}\cup\{(r+G_1)^2+G_2^2\le h\}]\le\Pr[\{G_2^2\le t\}\cup\{(r+G_1)^2+G_2^2\le h\}]. $$
Motivation:
I formulated the inequality in…

Yihan Zhang

- 153
- 7

**7**

votes

**2**answers

### Evaluating $ \int_{-\infty}^{t} e^{-(\tau+a)^2} \mathrm{erf}(\tau) \mathrm{d}\tau$

I need to evaluate this integral:
$$
I(t,a) = \int_{-\infty}^{t} e^{-(\tau+a)^2} \mathrm{erf}(\tau) \ \mathrm{d}\tau
$$
where the $\mathrm{erf}(\tau)$ is the error function.
I can prove that this integral converges. By employing the python…

Ghoti

- 103
- 7

**7**

votes

**1**answer

### Is a Gaussian Processes equivalent to a linear transformation of itself?

I am wondering whether a non-degenerate continuous Gaussian process is equivalent in distribution to a linear transformation of itself.
More specifically, let $T$ be a separable, complete and compact metric space, $C(T,\mathbb{R})$ the set of…

Peter

- 632
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**6**

votes

**1**answer

### Solution of SDE $dX(t)=a(t)dt+b(t)dW(t)$ is gaussian?

A stochastic process $X(t)$ by definition is gaussian iff all its finite-dimensional joint probability density functions are multivariate gaussian. Namely iff given the times $(t_1,t_2,...,t_n)$ , the random variables $(X(t_1),X(t_2),...,X(t_n))$…

Antonio19932806

- 205
- 1
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**6**

votes

**1**answer

### Fake Brownian motion - not Gaussian

Let $G$ be a standard normal random variable and define two standard Brownian motions $(W_t)_{t \ge 0}$, $\&$ $(B_t)_{t \ge 0}$. Assume $G, (B_t)$ and $(W_t)$ are independent.
Moreover, define that process $Y_t$ by
$$
Y_t =
\begin{cases}
B_t, & 0…

dorispadilla1221

- 559
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**6**

votes

**1**answer

### Convolution of tempered distribution($K$) and gaussian. if $K = K*e^{-\pi |x|^2}$, then $K$ is first degree polynomial.

Q : I need to prove that if $K$ is tempered distribution on $\mathbb{R}$ satisfying:
\begin{equation}
K = K*e^{-\pi |x|^2}
\end{equation}
then $K$ is first degree polynomial. mean $K(x) = Ax + b$
Remark: The question was changed. The original was…

shestak

- 95
- 7

**5**

votes

**0**answers

### Farmer wants to know how wet their field is

Problem
A farmer wants a better understanding of rainfall on their field. Assuming rain falls randomly and with equal likelihood over the entire field, the farmer thinks they can model the volume of rainfall $R$ on any given patch as normally…

Greedo

- 191
- 6

**5**

votes

**0**answers

### Why should the sum of squares of two Independent normals be memory-less

In section 11.3.1 of Introduction to probability models by Ross (10th edition), a very strange phenomenon is described. If you take two independent standard normal distributions and sum their squares, you get an exponential distribution with rate…

Rohit Pandey

- 5,457
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- 44

**5**

votes

**2**answers

### Chi Squared Clarification

Suppose you have $Z_1, Z_2, Z_3$ which are all independent standard Gaussian variables.
Suppose you have
$$A=\frac{(Z_1-2Z_2+Z_3)^2}{12}+\frac{(Z_1-Z_3)^2}{4}+\frac{(Z_1-Z_2)^2}{4}+\frac{(Z_1+Z_2-2Z_3)^2}{12}.$$
$A$ is $\chi_2^2$, but I don't see…

jeff123

- 397
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**5**

votes

**4**answers

### Intuitive explanation for $\lim\limits_{n\to\infty}\left(\frac{\left(n!\right)^{2}}{\left(n-x\right)!\left(n+x\right)!}\right)^{n}=e^{-x^2}$

In this post I noticed (at first numerically) that:
$$\lim\limits_{n\to\infty}\left(\frac{\left(n!\right)^{2}}{\left(n-x\right)!\left(n+x\right)!}\right)^{n}=e^{-x^2}$$
This can be proved by looking at the Taylor…

tyobrien

- 3,195
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**5**

votes

**0**answers

### Analyticity of determinant formula for Gaussian integral

It is a well known fact that $\int_{\mathbb{R}^n} e^{-\frac{1}{2}x \cdot A x} dx = \sqrt{\frac{(2\pi)^n}{\det{A}}}$ for real, positive definite $A$. The left hand side of the equation make sense for any complex-symmetric $A$ with real part positive…

UtilityMaximiser

- 928
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**4**

votes

**1**answer

### Comparing the Training Costs of Machine Learning Algorithm: A Mathematical Perspective

Recently, I was looking at the optimization functions required in training Kernel Based Methods compared to Neural Networks.
1) Kernel Methods:
For instance, I was looking at the optimization in Support Vector Machines:
And Gaussian Process…

stats_noob

- 1,696
- 8
- 15

**4**

votes

**1**answer

### Positive semi-definitness of modified RBF Kernel

Let's say I have an imaging space made of 100x100 pixels and I want to make a covariance matrix using RBF kernel (Gaussian kernel). In other words, say the covariance matrix is $C\in\mathbf R^{10000\times10000}$, then the $i,j$th element is…

JAEWON LEE

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