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
559 views

Lognormal definition. Clarity please?

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
0
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1answer
25 views

Stats Normal distribution [on hold]

A company pays it employees an average wage of $\$15.90$ an hour with standard deviation $\$1.50$ per hour. Assume the wages are approximately normally distributed. a) What proportion of employees ...
0
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0answers
19 views

What is the superior strategy in opening cells here?

This is a problem I just completed for a quiz online. Was wondering if someone could help me reason this out. Suppose you receive a message and you are trying to decide whether Team Rocket or Team ...
-4
votes
1answer
80 views

Prove that if $X \sim N(\mu, \sigma^2)$, then $X \sim \mu + \sigma N(0, 1)$ [closed]

As above. Also how is the general case proved for multivariate Gaussian? edit: I'm not sure why people voted to put this on-hold, it's just asking for a justification of a commonly used statistical ...
-2
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0answers
11 views

normal dribution -with mean time and standard deviation [on hold]

The time taken by passengers arriving at an airport has been recorded as follows: The mean walk time from the chocks on of an aircraft that lands to the immigration counter is 5 mts, with a standard ...
3
votes
1answer
304 views

How to plot standard deviation rings of a bivariate normal distribution?

I'm working on a project right now where I have Gaussian distributions, and I want to create a graphic that represents them. I'm not sure how to generate the ellipse that represents say 1 standard ...
-3
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0answers
12 views

To show that if T is a Weibull then aT also have a Weibull distribution [on hold]

(a)Show that if $T$ is a Weibull then $aT$ also have a Weibull distribution (b)$S(aT)$ is also Weibull Survival function
2
votes
1answer
54 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
1
vote
1answer
28 views

Estimate variance, how to find expected value of $x^2 [n]$

We have data $x_0, x_1, \ldots, x_{N-1}$ where the $x_n$'s are independent and identically distributed as ${\rm Normal}(0,\sigma^2)$. The estimate of $\sigma^2$ is $$\hat \sigma^2 = \frac{1}{N} ...
4
votes
2answers
221 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 ...
1
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2answers
250 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
0
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2answers
26 views

Area under Normal Distribution Curve

What is the formula that determines the Z-score table? More specifically, what formula can be used the equate the area underneath the normal distribution curve, without using the table?
0
votes
1answer
20 views

Confidence Interval for Regression Coefficient ,$\beta$

In the book 'Applied regression Analysis' by Draper/Smith, it is written that : Obtain individual $100(1-\alpha)\%$ confidence interval for the various parameters separately from the formula ...
0
votes
2answers
98 views

Proof of the affine property of normal distribution for a landscape matrix

The widely used/mentioned/assumed affine property of multivariate normal distributions says that: Given a random vector $x \in R^N$ with a multivariate normal distribution -- $x \sim N_x(\mu_x, ...
1
vote
1answer
37 views

Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
2
votes
1answer
37 views

Generating a nonrandom sequence which has a normal distributed density

I need to create an algorithm in a computer program (Fortran90) which generates a sequence of $n$ (between $10$ and $10^6$) numbers $z$ that follow a normal distribution. Restrictions: Has to ...
6
votes
4answers
97 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 $E(X^2)=1$?
0
votes
1answer
27 views

Mantel-Haenszel $\chi_1^2$ statistic

I was doing a particular example from the book Epidemiologic Research by Kleinbaum(example 15.6) and didn't understood some basic statistical aspect. ...
0
votes
3answers
38 views

Question about normal approximation and variance

This isn't so much a question about getting a right answer as much as it's about understanding a mathematical concept, but I will give you the problem that spawned it: An analysis of data shows that ...
1
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0answers
61 views

Probability:questions on characteristic functions

A well-known example to show that two random variables whose marginal distributions are normal, do not need necessarily be jointly normal is achieved by letting $X, Y $ have the following joint ...
0
votes
1answer
19 views

Plotting Normal Distribution using Excel

I was trying to experiment some stuff (scaling issues and hypothesis testing) with normal distribution. While doing so, I found out that : NORM.S.DIST(0, FALSE), which takes Z-value, returns prob. ...
0
votes
1answer
59 views

integral with pdf of a gaussian

$$ I = \int_{0}^{\infty} x \phi(x) dx $$ where $\phi(x)$ is the pdf of a normal distribution. Here I read that: If $X = \mu + \sigma U$ with $U$ a std normal, $$ I = E[\mu + \sigma U; mu + \sigma ...
0
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0answers
19 views

User of a System

Given a system with n users and each user will only use the system once (for an hour) during a year. The user will only access the system during business hours (so ...
3
votes
2answers
62 views

$\frac{1}{\sqrt{2\pi}}\int_\frac {1}{2}^0\exp(-x^2/2)dx$

How do we analytically evaluate $J=\frac{1}{\sqrt{2\pi}}\int_\frac {-1}{2}^0\exp(-x^2/2)dx$? This is what I tried: $$ J^2=\frac{1}{{2\pi}}\int_\frac {-1}{2}^0\int_\frac {-1}{2}^0\exp(-(x^2+y^2)/2)dxdy ...
1
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5answers
2k views

The sum of n independent normal random variables.

How can I prove that the sum of $X_1, X_2, \ldots,X_n$ random variables, all of which have normal distributions $N(\mu_i, \sigma_i)$, is a random variable that is itself normally distributed with mean ...
1
vote
1answer
48 views

How to fill in these steps to evaluate this Gaussian integral?

As a part of a much bigger problem, I came across this integral $$\int_{-\infty}^{\infty}\ln(|x|)\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}dx$$ which represents ...
0
votes
0answers
12 views

Combining independent Gaussian probabilities

I am using three Gaussian distributions with which I generate random numbers to represent many candidate xyz points. I use some selection criteria (details not particularly relevant) to decide on ...
3
votes
0answers
43 views

On integration of a Gaussian-like function over a region $g(\mathbf{x})\leq 1$

Let $X$ be a random variable which follows an $n$-dimensional Gaussian distribution with mean vector $\mu\in\mathbb{R}^n$ and covariance matrix the symmetric positive definite $n\times n$ matrix ...
2
votes
1answer
94 views

Is there an analytical solution to Gaussian integral $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx$?

I wonder if there is an analytical solution to $$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx,$$ where $a, b>0$. I know, of course, that the antiderivative of the fraction is a version ...
2
votes
1answer
23 views

Marginalization of a paramter in Gaussian

If $\theta \sim N(\mu,\sigma_o^2)$ and $\mu \sim N(0, \sigma_1^2)$ what is the marginalized $P(\theta)$. Is it $N(0,\sigma_o^2+\sigma_1^2)$?
-5
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0answers
45 views

What's the value of Epsilon?

This is one solution, which I'm learning on. I don't know why ε is 0,02.
0
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1answer
10 views

Distribute range of score among objects

I need some help with the following. I have 10 or X amount of subjects with a rating and would like to distribute a score of 1 to 5 between them based on their rating. The subject with the highest ...
0
votes
1answer
530 views

Distribution of the sum of squared independent normal random variables.

The sum of $k$ independent standard normal random variables $\sim\chi^2_k$ I read here that if I have $k$ i.i.d normal random variables where $X_i\sim\mathcal{N}(0,\sigma^2)$ then ...
1
vote
1answer
11 views

Will statistical analysis of transformed data hold for the original one?

I have a data with distribution like chisq-squared one. But ANOVA and t-test need the data to be normal distributed. So I want to do the Box-cox transformation to the data, but my concern is will the ...
1
vote
1answer
47 views

Extreme Value Theory - Show: Normal to Gumbel

The Maximum of $X_1,\dots,X_n. \sim$ i.i.d. Standardnormals converges to the Standard Gumbel Distribution according to Extreme Value Theory. How can we show that? We have $$P(\max X_i \leq x) = ...
0
votes
1answer
59 views

Given $f_X$. Integrate $\int_0^\infty \log_2 (x+1) f_X \, dx$.

Say $Y=Log_2[1+x]=g(X)$ and $f_X = \frac{e^{-\frac{(\mu -\log (x))^2}{2 \sigma ^2}}}{\sqrt{2 \pi } x \sigma }$ is Log-normal density function: [Wiki] Find E[Y]? Since $E[Y] = \int_0^\infty y f_Y \ ...
0
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1answer
27 views

Normal distribution Z score

Problem: The observed error "E" in a series of measurements is normally distributed with mean of 0. Approximately 2% of error are -10 or less. Approximately what fraction of the measurements have ...
2
votes
2answers
83 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
2
votes
1answer
30 views

Show that $d^T Z\sim N(d^T\mu, d^TVd)$ [duplicate]

Consider $Z=(Z_1,\ldots,Z_n)^T\sim N(\mu,V)$ with $\mu=(\mu_1,\ldots,\mu_n)^T$ and $V=\text{Cov}(Z)$. Show that for $d\in\mathbb{R}^n$ it is $$ d^TZ\sim N(d^T\mu,d^TVd). $$ For me it ...
2
votes
1answer
24 views

Gaussian prior favors values closest to zero?

I am reading an article on Bayesian Logistic Regression, where they're using Logistic Regression, imposing a Gaussian prior (with mean = 0) on its parameters. They state that a Gaussian prior favors ...
1
vote
1answer
51 views

Determining whether random variables are independent

If I have two random variables as follows: 1) A Gaussian distribution of wifi signal strengths at a known point 2) A Gaussian distribution of wifi signal strengths at an unknown point (Note that ...
1
vote
3answers
250 views

Scaling the normal distribution?

I might just be slow (or too drunk), but I'm seeing a conflict in the equations for adding two normals and scaling a normal. According to page 2 of this, if $X_1 \sim N(\mu_1,\sigma_1^2)$ and $X_2 ...
1
vote
1answer
27 views

Show $\lim_{n\to\infty}\sqrt{n}\bigg(\frac{\sum_{j=1}^{n}X_j}{\sum_{j=1}^{n}X_j^2}\bigg)=Z$

Let $(X_j)_{j\ge 1}$ be independent, double exponential with parameter $1$. Show that; $$\displaystyle\lim_{n\to\infty}\sqrt{n}\bigg(\frac{\sum_{j=1}^{n}X_j}{\sum_{j=1}^{n}X_j^2}\bigg)=Z$$ where ...
0
votes
0answers
13 views

K Weighted Nearest Neighbour - Comparing Gaussians

This problem relates to a WiFi Indoor Positioning method - http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3812643/ The problem consists of the following steps: 1) In a database, we will have stored the ...
8
votes
1answer
272 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 ...
1
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0answers
16 views

Draw and compare the likelihood using R

The following shows the heart rate (in beats/minute) of a person, measured throughout the day: 73, 75, 84, 76, 93, 79, 85, 80, 76, 78, 80. Assume the data are an iid sample from ...
0
votes
1answer
27 views

How to calculate $\int\limits_{-\infty}^{+\infty} \frac{n-1}{\alpha}\Phi(\frac{x}{\alpha})^{n-2}\phi(\frac{x}{\alpha})^2dx$?

I was working on a research project that involves taking the integral of $$\frac{n-1}{\alpha}\int\limits_{-\infty}^{+\infty} ...
0
votes
1answer
38 views

Normalizing a dataset from the interval [0,1] with fixed properties.

So I have a rather large dataset where values are from the interval $[0,1] \in \mathbb{R}$. But the problem is that a big portion of the values are extremely close to $0$. So firstly I'm looking for ...
1
vote
1answer
23 views

Normal Distribution and Probability on Excel

The size of fish in a lake follows a Normal Distribution with mean m = 1 lb 4 oz and standard deviation s = 3 oz . Fish that weigh less than 1 lb 9 oz must be released back into the lake. Bill ...
1
vote
1answer
31 views

Given a histogram, programatically, how do I find the normal distributions that comprise it?

I will be getting data in at around 100 frames per second, and I need to compute the normal distributions that comprise a set of 48 data points. The distributions can partially overlap, but will ...