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

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1answer
20 views

Efficient methods for drawing random numbers and Monte Carlo for Tsallis q-Gaussians

I would like to draw random numbers from the q-Gaussian used in "Tsallis statistics." This is specifically the distribution $$ f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) $$ where $$ e_q(x) = [1+(...
0
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1answer
27 views

If $Y\sim N(\mu T,\sigma^2 T)$ then $Y=\mu T+\sigma \sqrt{T} Z$ where $Z\sim N(0,1)$

Let $S_T$ be the price of a traded asset at time $T$. Also let: $\ln(\frac{S_T}{S_0}) \sim N(\mu T,\sigma^2 T)$ My question is, how is it that: $\ln(\frac{S_T}{S_0})=\mu T+\sigma \sqrt{T} Z$ where $...
2
votes
1answer
66 views

The expected value of an order statistic for normal random variables

Let $X_1$ and $X_2$ be a random sample from normal distribution with mean equal to zero and variance $\sigma^2$. Prove $E[X_{(1)}]= \frac{-\sigma}{\sqrt{\pi}}$. May I have to standarize the sample? ...
0
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1answer
21 views

Conditional gaussians, particular calculation

I'm looking for confirmation that my solution to this problem is correct. The result seems unintuitive. Given $\{X_i\}_{i = 1}^{10}$ $0$ mean jointly gaussian RVs with $\mathbb{E}X_iX_j = 2^{-|i - j|...
2
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1answer
34 views

$2^\text{nd}$ Derivative of normal distribution, evaluated at one standard deviation

What is the $2^{nd}$ derivative of the normal distribution at one standard deviation? The normal distribution is given by $N(x,\mu ,\sigma)=\frac{1}{\sigma\sqrt{2\pi }}e^{-\frac{(x-\mu)^2}{2\sigma^2}}...
1
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1answer
39 views

Triangle whose corners are N(0,1) variables

A friend of mine and I have been exchanging and solving math puzzles and this is the last one: A triangle is formed by three points on a plane, whose $x$ and $y$ coordinates are $N(0,1)$ random ...
1
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0answers
17 views

What is the expected number of extreme points in a set of points drawn from a normal distribution?

Say I construct a set of points $S$ by drawing $k$ points from $\mathbb{R}^{n}$ independently from say a gaussian distribution*. I am wondering how the expected size of the set $E$ consisting of the ...
0
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1answer
51 views

Finding parameters of normal distribution

I am trying to solve this problem: A factory produces batteries whose duration in hours for a particular has a normal distribution with $\mu_0=53$ and ${\sigma_0}^2=25$. Now suppose that there is a ...
0
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2answers
67 views

Expectation of the inverse of a shifted squared of a normal random variable!

I am trying to calculate $\int_{-\infty}^{\infty} \frac{p(x)}{x^2+\epsilon}dx$ where $p(x)$ is the density function of a normal random variable. Numerical experiments easily show that it is nothing ...
3
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1answer
48 views

Fourier transform invariant functions other than the bell curve? [closed]

Are there any functions that are their own Fourier transforms other than $e^{-\pi x^2} $?
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0answers
41 views

Cumulative distribution function of function of normal random variables

I have $X_1 \sim N(0, 4), X_2 \sim N(0, 4), X_3 \sim N(3, 1), X_4 \sim N(1, 9)$, they are independent. I need to find cumulative distribution function of $\xi = \frac{X_3X_2 + X_4X_1}{ \sqrt{X_4^2 + ...
0
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2answers
66 views

What is the proportion of the population listed below is highly advanced (greater than or equal to 145?)

Intelligence quotients (IQs) measured on the Standford Revision of the Binet-Simon Intelligence Scale are normally distributed with a mean of 100 and a standard deviation of 16. ...
1
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0answers
22 views

a puzzling result for quadratic ratio of normals

Note: Below I use $\approx$ to indicate equivalence in distributions and use $\sim$ to indicate that how the distribution law of a certain variable is defined. Let $X_{1,i}$ and $X_{2,i}$ be two ...
0
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0answers
18 views

Probability normal distribution; changing the SD.

A company manufactures tennis balls whose diameters are normally distributed with mean $67$ mm and standard deviation $1$ mm. The manufacturer wants to change the standard deviation so that $99$...
2
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1answer
569 views

How can I solve the probability of a sample mean exceeds population mean if I'm not provided means?

The question states "Times spent studying by students the week before exams follows a normal distribution with standard deviation of 8 hours. A random sample of four students was taken in order to ...
3
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1answer
63 views

Distribution of $Y= \frac{X_1}{|X_2|}$?

If $X_1$ and $X_2$ are independent and identically distributed Gaussian random variables with parameters $0$ and $\sigma^2$, how do I find the distribution of $Y= \frac{X_1}{|X_2|}$? I'm not supposed ...
1
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0answers
36 views

Bayesian Gaussian Mixture model

I am trying to fit basic Gaussian mixture with a Bayesian model. My likelihood function is Gaussian, with std=1, and the only parameter is the mean, chosen from $\{0,1,\dots,14,15\}$ and my prior is ...
4
votes
1answer
126 views

Find expectation of Z (normal)

Assume the following PDF. $$f(z) = \frac{1}{\sqrt{2π}}e^{-(z^2 / 2)}$$ Find E(z). For now, I got E(z) = $\int_{-\infty}^{\infty} f_z(z) dz $ And for any odd function F, (i.e. F(−x) = −F(x)∀x ∈ R),...
0
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2answers
28 views

Distribution of $Y$ derived from standard normal

If $X$ is $ N(0,1)$, then what the distribution of $Y$, where $Y=X$ when $|X|\leq 1$ and $Y=-X$ when $|X| > 1$. My attempt: when $|Y| \leq 1$, $F(y) = \Phi (y) $. When $|Y| > 1$, $F(y) = 1 - \...
1
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0answers
12 views

Multi sample chi-square distribution

I have a question from my assignment. For a total of 150 households selected randomly from 3 cities -- A, B and C -- of 50 each, the sample mean and covariance matrices are as below: For A, ...
0
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0answers
36 views

How to prove that the normal pdf function satisfies the forward Kolmogorov equation

How to prove that the normal pdf function $ P(x,t)=\frac{1}{\sqrt{2\pi\sigma^2t}}e^{-\frac{(x-\mu t)^2}{2\sigma^2t}}$ satisfies the forward Kolmogorov equation $ \frac{1}{2}\sigma^2\frac{\partial^2P}{\...
0
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0answers
50 views

What is the correlation of two normal distributions with equal mean and known relation between variance

If I take the sample mean of a scaled $\chi$ distribution of $N$ samples, the distributions of these means should lead to a normal distribution $\mathcal{N}(\mu,\sigma)$. This procedure is repeated $M$...
0
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1answer
385 views

Help for convolution of two Multivariate Gaussian PDFs

I am looking for a proof for convolution of two multivariate Gaussians (where each Gaussian has multi-dimensional mean and co-variance). I found a proof in here: http://www.tina-vision.net/docs/memos/...
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0answers
16 views

Question on Continuity Correction for Hypothesis Testing

This is a question on continuity correction in hypothesis testing for the approximation of Binomial to Normal Distribution: (i) Deng wishes to test whether a certain coin is biased so that it is ...
4
votes
1answer
413 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 ...
1
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1answer
122 views

Can I plot a normal probability distribution given the number of trials, average, minimum, maximum, and standard deviation?

I have the information about processing timing of db transactions. I was wondering if the info I have is sufficient to plot a normal probability distribution graph. Data available: 560000 hits ...
-4
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1answer
86 views

President Obama proposed the elimination of taxes on dividends paid to shareholders on the grounds that they result in double taxation.

President Obama proposed the elimination of taxes on dividends paid to shareholders on the grounds that they result in double taxation. The earnings used to pay dividends are already taxed to the ...
1
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0answers
42 views

The distribution of gaussian 2D vector given the distribution of average

My question: Let $X_1, X_2\sim N(\theta,1)$. Let $\bar X = aX_1+(1-a)X_2$ for $0<a<1$. Find the distribution of $(X_1,X_2)$ given $\bar X$. Update: My previous try is wrong. I delete it. ...
0
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1answer
16 views

Maximum Density in Normal Probability Density where x has been replaced with $\mu+y^{(1/2)}$

Thanks for reading. Let's say you have the normal probability density: $P(x|\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}$ Now, if you set $ y = (x-\mu)^2$, what's the ...
-4
votes
1answer
59 views

Sufficient statistic for $N(\theta,\theta^2)$ [closed]

Let $X_1,\ldots,X_n$ be a randon sample of the normal distribution with parameters $(\theta,\theta^2)$ How can I find a sufficient statistic for $\theta$? Is there an easy way to do it?
0
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1answer
7 views

Is this proof of the formula of the covariance of linearly transformed random variables correct?

I want to prove that $cov(AX+a,BY+b) = Acov(X,Y)B^t$ for any matrices $A,B$ and vectors $a,b$ if $X$ and $Y$ are normal distributed vectors, where $m$ is the mean of $X$ and $n$ the mean of $Y$ Is ...
0
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0answers
16 views

Minimization of vector norm in the reals

Let $V(x)$ be some vector whose values depend on $x$. That is, $$V(x) = \begin{pmatrix} v_1(x) \\ v_2(x) \\ \vdots \\v_n(x) \end{pmatrix}$$ How can one solve the following equation in the reals?: $$\...
0
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1answer
51 views

Why is a $2\pi$ in the gaussian distribution function?

I am using a 3 dimensional gaussian point spread function in the form of $$\frac{1}{\sqrt{(2\pi)^3}\sigma^3}e^{-\frac{r^2}{2\sigma^2}}$$ being $r^2$ the square of the distances $x^2 + y^2 + z^2$, to ...
0
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2answers
1k views

Calculating percentile value from mean and standard deviation of a normal distribution

I have to write some code to calculate the 95th percentile from a databaset which is normally distributed. I am easily able to calculate the mean and the standard deviation, which define the ...
2
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1answer
50 views

How to show this function is increasing? Related to Normal distribution.

Numerically, it seems the following function $F(x)$ is increasing in $x$. How can I show it analytically? $$F(x)=G(x)L'(x)$$ where $L(x)=\frac{(1-G(x))^3}{G'(x)}$ and $G(x)=\int_{-\infty}^x \frac{e^{...
3
votes
1answer
72 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 ...
2
votes
1answer
22 views

How to calculate the best step for a prices range of multiple products' prices?

Assume that I have N products with thier prices. ${P1, P2, P3, ..., Pn}$. I have the maximum price $Pmax$ and the minimum price $Pmin$ . I want to calculate the best steps to quantize this interval ...
0
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0answers
26 views

Why does |z| > 0,35 give an answer for the middle of the standard Normal model, instead of the tails?

I have a question in a textbook. It reads like: Assuming a standard Normal model, what is the probability for each of the following cases? And the case is: |z| > 0,35 So I took this to ...
0
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0answers
16 views

Variance of the product of random vector $Z\cdot Z^T$

Suppose that $Z$ is $n\times 1$ random vector following multivariate normal distribution with mean $\mu$ and covariance matrix $\Sigma$. Any ideas to find $Var(Z\cdot Z^T)$? It seems that it is very ...
0
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0answers
36 views

Numbers not distributed evenly

We have a large (2000+) set of numbers that we are assigning grades to. This is the logic But no matter how many different data sets I use, grades fall into A, C, or F, i.e. I understand that ...
3
votes
1answer
126 views

Expected maximum absolute value of $n$ iid standard Gaussians?

I have a problem where my errors are normally distributed and I want to know what the expected maximum error is if I repeat the process $n$ times. What is the smallest constant $C$ such that the ...
1
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3answers
76 views

Distinguish Normal Distribution, Gaussian Distribution and Normalised Gaussian Distribution?

As I understood it, the 'normal distribution' is $$\frac{1}{\sqrt{2\pi}}\exp\left(\frac{-(x-\mu)^2}{2{\sigma}^2}\right)$$ Now according to this the 'normal probability density function' is $$f(x)=\...
0
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0answers
38 views

Semi Log-scaled Density Plot

I have the following plot, which represents the density of articles' length per gender. The figure is generated using ggplot2 based on 3000 Male articles (i.e., articles about famous male ...
2
votes
1answer
49 views

How to arrive at $\int x^2 \phi(x) \, \mathrm{dx} = \Phi(x) - x\phi(x) + C$

I found the following result in Wikipedia $\int x^2 \phi(x) \, \mathrm{dx} = \Phi(x) - x\phi(x) + C$ where $$\phi(x) = \frac{1}{(2\pi)^{1/2}} \mathrm{e}^{-\frac{1}{2}x^2}~\mathrm{and}~\Phi(x) = ...
1
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0answers
28 views

Distribution of conditional expected value

Motivated from an application in economics, I would be interested in a simplified solution for $C$, where \begin{equation*} C = E\bigg[exp \bigg\{-\frac{1}{2} \frac{(E[\theta|S]-a)^2}{b}\bigg\}\bigg] \...
0
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0answers
14 views

Independent sampling and joint normality

I know that random variables which are each normally distributed and independent have a joint normal distribution. However, given that I have normally distributed random variables, but no knowledge ...
8
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2answers
125 views

Multivariate Normal Distribution: Relationship between two conditional probabilities.

Suppose I have a multivariate normal random variable $Z$ which has $n$ dimensions. Suppose I have a vector $x$. Set $i$ as a number between $1$ and $n$ and $k$ as a number between $1$ and $n-1$. Can ...
1
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0answers
57 views

If $0< 2\varepsilon < \sigma^2 < 1$ then $\prod\limits_{i = 1}^n (1 + \varepsilon + \sigma \xi _i )$ converges almost surely to $0$

I posted this question a few days ago and there were some errors in my post. I have fixed them and it should be all right now. Hope someone can help with my confusion. Let $(X_n)_{n\ge0}$ denote an ...
2
votes
1answer
86 views

Find all solutions of $f$ such that $ \operatorname{Cov}(f(x),x)=c $ where $x \sim N(\mu,\sigma^2)$ and $c$ does not depend on $(\mu,\sigma^2)$

Find all solutions of $f$ such that $$ \operatorname{Cov}(f(x),x)=c$$ where $x \sim N(\mu,\sigma^2)$ and $c$ does not depend on $(\mu,\sigma^2)$ I think the only solution is $f$ is constant (almost ...
1
vote
1answer
47 views

Normal Distribution Application

Given: $\mu=80$, $\sigma=15$, $500$ respondents a. Find $P(74\lt x \lt 101)$ b. Find number of respondents with score $\lt 98$ For a., my answer is $0.5746$ (using the formula $z=\cfrac{x-\mu}{\...