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

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0
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1answer
13 views

generate random number from normal distribution

Can any one explain in which range I am going to get random numbers, if I was said generate random number from normal distribution with mean=50 and std_dev=25, what does it exactly means..I tried to ...
0
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0answers
4 views

Variance reduction factor using gaussian filtering

I am currently trying to find the variance reduction ratio using gaussian filtering. For a simpler filter (as mean filtering for example), I am able to calculate it easily to find the well known ...
0
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1answer
8 views

Modelling a normal-like single-ended random variable

I am trying to model a of (normal-distribution-like) discrete random variable using the normal distribution. This is what I understand so far: First, I approximate the mean of the normal ...
0
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1answer
54 views

Getting a p-value from a histogram?

A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and ...
0
votes
1answer
17 views

Regression when the variance of the residuals depends on the independent variable

When the residuals follow a normal distribution, the most likely function that fits the data is found using least squares. In that case: $y = f(x_i) + r_i, \quad r\sim\mathcal{N}(0, \sigma^2)$ ...
0
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0answers
41 views

Compound Gaussian distribution

Let $\mathbf{a},\mathbf{b}\sim \mathcal{N}(\mathbf{0},\sigma^2\mathbb{I})$ and let $A$ be the circulant matrix defined to have $\mathbf{a}$ as its first column. I'm trying to study the behaviour of ...
2
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1answer
24 views

Minimum number of samples to take so that proportion of smokers in sample is within a certain threshold?

What is the minimum number of random samples that should be taken so that with probability at least 0.95, the proportion of smokers in the sample will not differ from the unknown population of smokers ...
1
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0answers
7 views

Finding posterior of normal distributions and logistic regression.

$P(w_0 | x) = \frac{1}{1 + e^{-log\frac{P(x|w_0)}{P(x|w_1)}-log\frac{P(w_0)}{P(w_1)}}}$ Note: x = $[x_1, \dots, x_d]^T$; a $d$ dimensional vector. $w$ can take on one of two values: $w_0$ or $w_1$. ...
0
votes
1answer
18 views

Distribution combinations

How many ways can 25 identical pencils be distributed between two people?.Each all pencils must be shared out. A) Each person must have at least 5 pencils? B) Each person must have at least 7 ...
1
vote
1answer
45 views

Sum of dependent normal random variables

Let ${\bf X} =(X_1,\ldots,X_n)'$ be a vector of random variables that may be dependent and let ${\bf a}=(a_1,\ldots,a_n)'$ and ${\bf b}=(b_1,\ldots,b_n)'$ be nonrandom vectors with $a_i \neq 0$ and ...
0
votes
1answer
21 views

Proving some properties about the expected first order statistic (maximum) with respect to sample size.

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as ...
3
votes
2answers
48 views

What is the reason for the one-half in the normal pdf's gaussian (i.e. : why $\exp(-x^{2}/2)$ instead of $\exp(-x^{2})$ )

It doesn't seem to relate to normalization, as the normalizing constant adapts to every possible "upstairs formulation", and in the standard case is $\displaystyle\frac{1}{\sqrt{2\pi}}$. Does it ...
1
vote
1answer
67 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
-3
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0answers
9 views

Proof of “Normal approximation to the log-normal distribution” [closed]

I saw the post about the normal approximation to lognormal (Normal approximation to the log-normal distribution). The proof is shown as well. Yet as I'm looking for the proof in a journal article form ...
3
votes
4answers
226 views

$\int_{0}^{\infty}xe^{-x^2/2}dx= 1$?

$X \sim N(0, 1)$ $$E(|X|) = \frac1{\sqrt{2\pi}}\int_{-\infty}^{\infty}|x|e^{-x^2/2}dx= \frac{2}{\sqrt{2\pi}}\int_{0}^{\infty}xe^{-x^2/2}dx=\sqrt{\frac{2}{\pi}}$$ I don't understand how the last ...
2
votes
1answer
22 views

Surjectiveness of standard-normal c.d.f. [closed]

Let $\phi:\mathbb R \to (0,1)$ be a function defined as $\phi(y)=\int_{-\infty}^y\dfrac{1}{\sqrt{2\pi}}e^{-\dfrac {x^2}{2}}dx , \forall y\in \mathbb R$ , then is it true that $\phi$ is surjective ? If ...
0
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2answers
24 views

Normal Distribution,standard deviation and probability question.

According a study, the duration of a match in World Cup is approximate normally distributed with the mean 111 minutes and standard deviation 5 minutes (including the break between the halves). ...
-1
votes
1answer
29 views

Stats Normal distribution [closed]

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 ...
2
votes
2answers
85 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 ($\text{W}$). ...
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
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1answer
30 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 ...
1
vote
1answer
40 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 $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
-4
votes
1answer
81 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
votes
1answer
41 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 ...
0
votes
1answer
32 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
43 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 ...
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
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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
63 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 ...
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0answers
65 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 ...
1
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1answer
50 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
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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
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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
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 $N(0,\sigma_o^2+\sigma_1^2)$?
2
votes
1answer
100 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 ...
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 ...
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votes
0answers
46 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.
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 ...
0
votes
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
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 ...
0
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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 \ ...
1
vote
1answer
50 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) = ...
1
vote
1answer
28 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
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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 ...
1
vote
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
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1answer
29 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} ...
1
vote
1answer
53 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
1answer
32 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 ...
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 ...