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

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1
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2answers
70 views

Statistical Problem (part 2)

Following my question I found another problem. Having the same data from the other question: There are 2 melon stores. The melon weights follow a normal distribution. Store A -> μ = ...
0
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2answers
110 views

Normal distribution, statistical problem

Before proceeding to the question, bear in english is not my native language and therefore technical terms may be wrong. So, I'm trying to solve the old exam question, and I have different results ...
1
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1answer
194 views

Well known probability distributions defined on a $n$-dimensional simplex besides the Dirichlet distribution?

Are there well known probability distributions defined on a n-dimensional simplex besides the Dirichlet distribution where the variation of of each component doesn't vary as much when the mean of the ...
0
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0answers
44 views

Cauchy Random Variable Question [duplicate]

Possible Duplicate: How calculate the probability density function of $Z = X_1/X_2$ Need help with the following problem: Suppose that X and Y are independent normal random variables with ...
1
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2answers
1k views

If 100% of data fall within first two standard deviations of the mean, is the distribution Normal?

The empirical rule for a normal distribution suggests that 68% of data will fall within the first standard deviation, 95% within the first two standard deviations, and 99.7% will fall within the first ...
1
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0answers
228 views

Cramer’s decomposition theorem - find normal distributions within a normal distribution.

I know that Cramer's decomposition theorem says that any normal distribution can be expressed as the sum of multiple normal distributions. I have been searching for a method to divide a data set that ...
6
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3answers
224 views

How do I evaluate $\int \limits_{-\infty}^{a} e^{−t^2}dt$?

I know that $$I \equiv \int \limits_{-\infty}^\infty e^{−t^2} \, dt=\sqrt{\pi},\text{ and }\int \limits_{-\infty}^0 e^{−t^2} \, dt=\frac{\sqrt{\pi}}{2}.$$ However, I don't understand if (or how) I ...
0
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2answers
1k views

Adjusting mean and standard deviation

There's a set of 8 bags with the following weights in grams given: 1013, 997, 1013, 1013, 1004, 985, 991, 997 The mean is 1001.625, unbiased standard deviation is 10.86. I have the following ...
5
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4answers
5k views

Asymmetric Normal Probability Distribution

I'm looking for a continuous probability distribution a little bit like the normal distribution but asymmetric. In my opinion this distribution applies to phenomenons related to response time in ...
1
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0answers
91 views

continued fraction multivariate normal distribution?

After searching for a while, I wonder if a continued fraction representation exists for the multivariate Mills ratio $P(Z \geq x)/\phi_Z(x)$ where $Z \tilde\, N(\mu,\Sigma)$. The representation ...
3
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1answer
2k views

The variance in the average of a set of normally distributed random variables

I have a set of $M$ normally distributed random variables, $r_i$, each with an associated mean $u_i$, but the same variance $\sigma^2$. What is the variance of the average of these $M$ random ...
3
votes
1answer
321 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 ...
1
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1answer
630 views

conditional expectation of normal distribution using sigma algebra

Suppose $X$ and $I$ and independent, $X$ has a standard normal distribution and $I$ take values $1$ and $-1$ with equal probabilities. Let $Y = IX$. How would I find the distribution of $Y$ and ...
0
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0answers
66 views

Plot randomly oriented gaussian kernel

I would like to plot with scipy randomly oriented gaussian kernels. For a gaussian kernel along x and y axis (with an angle 0 w.r.t. coordinate system), I simply plot function ...
0
votes
3answers
117 views

Mixture of two mirror-image Gaussians

Suppose we are given a set of points $(x_i, y_i)\in\mathbb{R}^2$ and are told that they are drawn from a normal (Gaussian) distribution. It is a simple matter in that case to find the mean ...
3
votes
2answers
286 views

Two Sample Confidence Interval for Normal Distributions

Let's say I have two independent random samples $X_1, X_2, \dots, X_n$ and $Y_1, Y_2, \dots, Y_n$ from normal distributions with real, unknown means $\mu_x$ and $\mu_y$ and known standard deviations ...
1
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0answers
212 views

Polynomial approx to the Normal density

I have found several polynomial some approximations to the Normal CDF$^{(1)}$, but my question is: are there good polynomial approximations to the Normal PDF? Thanks $^{(1)}$ For example, some are ...
2
votes
1answer
220 views

How do I integrate this distribution?

I have a multinomial multivariate normal distribution of the form: $$\exp\left[-\frac{1}{2\sigma^2}(({\boldsymbol \beta}-\mu)^T\Sigma^{-1}({\boldsymbol\beta}-\mu)\right]$$ I wish to integrate with ...
1
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1answer
86 views

Is this a legitimate way to show that $[-ze^{-z^2/2}]_{-\infty}^{\infty}=0$? (Proving a statement about a function of a normal random variable)

The problem is, let Z be a standard normal variable and $n\geq1$ be an integer. Show that $E[Z^{n+1}]=nE[Z^{n-1}]$. Here's what I've got so far, miraculously: ...
4
votes
2answers
352 views

Property of gaussian integrals

Apologies if this has been asked before... I came across the following relation: if $$P(x_2, t_2 \mid x_1, t_1) = \frac{1}{\sqrt{2\pi\sigma^2(t_2-t_1)}}e^{-\frac{(x_2-x_1)^2}{2\sigma^2(t_2-t_1)}}$$ ...
0
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1answer
264 views

Question on statistics

I have a kind of weird question. But this wont be a harder one. Actually, i feel it is incomplete. I don't have much experience on statistics. But some advance user will be able to understand this ...
5
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0answers
77 views

Is there a way to exploit the fact that the covariance matrix has a blocked structure to more easily compute the multivariate normal density?

I'm trying to minimize the (negative) multivariate normal log likelihood (dropping constants): $$ \log |\boldsymbol\Sigma|\,+(\mathbf{x}-\boldsymbol\mu)^{\rm ...
4
votes
4answers
5k views

Calculate the expected value of $Y=e^X$ where $X \sim N(\mu, \sigma^2)$

I got a problem of calculating $E[e^X]$, where X follows a normal distribution $N(\mu, \sigma^2)$ of mean $\mu$ and standard deviation $\sigma$. I still got no clue how to solve it. Assume $Y=e^X$. ...
-2
votes
1answer
2k views

normal distribution and standard deviation

1.in a normal distribution data the standard deviation is greater than the quartile deviation and the mean deviation ?? in a normal distribution 31% of the items are under 45 and 8% are over 64 find ...
1
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2answers
95 views

Ranking students from 2 separate exams in single scale.

Is there a way to rank 2 student groups who face 2 separate exams in a single scale using z-score, given that there are enough student in each group to consider each score distribution a normal ...
0
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0answers
85 views

Modeling with a Bivariate Gaussian Distribution

In an earlier question I inquired about contour lines reflecting probability values in a Bivariate Gaussian Distribution. I have spent some time thinking and playing around with a Mathematica ...
5
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1answer
181 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? ...
2
votes
1answer
91 views

$\chi^2$ test and sampling variance

Let $f(x)$ denote the pdf of a $\chi^2$-distribution with $n\in\mathbb{N}$ degrees of freedom given by $$f(x) = \frac{2^{-n/2}}{\Gamma(n/2)}\cdot x^{n/2-1}\cdot\mathrm ...
0
votes
1answer
94 views

Normal Distribution Transformation

Suppose we have a normal distribution like $ f(x) = \mathcal{N}(\mu = 30, \sigma^2=10) $ and we transform it to another function by multiplying it to $ g(x) = 2x^2 $ the result would be: $ ...
2
votes
1answer
382 views

Closed Form of Normal Distribution

What does closed form in following sentence mean and why we need tables of c.d.f.? Normal distributions's p.d.f. cannot be integrated in closed form, and hence tables of the c.d.f. or computer ...
1
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0answers
39 views

Sample estimated normal distribution - what will be the expected effect of another sample?

Assume I already have n samples of a 2D variable. I can compute the sample mean and variance. If I assume that the samples are taken from a normal distribution, then using the mean and variance I get ...
2
votes
1answer
375 views

The distribution of uniformly random rotation of a i.i.d. Gaussian vector $\mathbf{x}$ given $\mathbf{x}$

Suppose that I have vector $\mathbf{x}$ that contains $n$ independently and identically distributed (i.i.d.) zero-mean Gaussian random variables $x_i\sim\mathcal{N}(0,\sigma^2)$. Also suppose I have ...
1
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1answer
578 views

prove a bivariate normal distribution

$X$ has a normal distribution. The conditional distribution of another random variable $Y$ given that $X=x$ is a normal distribution with mean $ax+b$ and variance $t^2$, where $a$, $b$, and $t^2$ are ...
0
votes
1answer
42 views

Normal distribution of the dosage

I am having an issue with the following task: For the dose of the MemPro medicine it is known that has normal distribution. Sample of 30 parameters has been taken into consideration and based on the ...
2
votes
1answer
227 views

Conditions for two normal R.Vs to satisfy bivariate normal distribution

I have to Normally distributed Random Variables X and Y which are correlated. What conditions should they satisfy so that their joint distribution is a bivariate Normal distribution?
0
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1answer
110 views

Confusion regarding autoregressive process

I was reading this article related to autoregressive processes of order $1$. According to wiki it is given by $$ x_t = \phi{x_{t-1}} + \epsilon \\ |\phi| < 1 \\ x_t|x_1,\ldots,x_{t-1} \sim ...
0
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1answer
87 views

$\alpha$-stable distributions and Gauss distribution tails differences.

I know that in a $\alpha$-stable distribution we have: $$ \lim_{x\rightarrow +\infty}f(x,\alpha,\beta)\sim -\alpha \gamma^\alpha \frac{\Gamma(\alpha)}{\pi}sin(\frac{\pi ...
1
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1answer
174 views

A simple question about normal distribution

Suppost we have a dataset as below: (Value,Frequency) pairs: (1,2), (2,4), (3,6), (4,8), (5,10) Can we say that this data is normally distributed, or have a normal distribution for this dataset?
1
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1answer
325 views

Combination of a normal r.v. with a log-normal one

It is well-known that a sum of normal r.v.'s is another normal r.v., and a sum of log-normal r.v.'s can be accurately approximated with a log-normal r.v. But what can we say if we have a mixture of ...
4
votes
1answer
4k 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 ...
3
votes
2answers
750 views

Examples of Student's T-distribution in real world empirical data?

I have recently stumbled onto some empirical (forecasting error) data that should be normally distributed. However, the normal distribution fits relatively poorly due to the abundance of data points ...
1
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1answer
97 views

Deriving the characteristic function for $N(0,2)$

Could someone please help me with an easy derivation of the characteristic function for a $N(0,2)$ distribution? Or a link to somewhere it is done.
1
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0answers
167 views

Combining general 1D normal distributions into a 2D distribution

My question is a generalization of the question asked here There is a point in 2-D space. I can measure the range of this point from two other locations. I get this measurement as a mean (range) and ...
4
votes
1answer
292 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 ...
2
votes
1answer
498 views

“Flattening” a 2D Normal Distribution

I would like to model the probability of a point being at a certain place on a 2D grid. The X coordinate of the point varies according to a normal distribution with mean $0$ and standard deviation ...
1
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1answer
676 views

Probability that one normal distribution is greater than the other when they are correlated i.e (Not Independent)

Would like to know how to approach this question: Probability that one normal distribution is greater than the other when they are correlated i.e (Not Independent). Seen the solution for the ...
1
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1answer
581 views

Constructing correlated random variables

How do I construct two correlated random variables with correlation $\rho$ given two i.i.d normal r.v.? Do I just multiply the correlation matrix by a vector generated with two i.i.d normal variables? ...
5
votes
1answer
210 views

Fractional Part of Sum of Sequence of Independent Normal Random Variables

I'm trying to prove that if $X_n$ iid normal $S_n = \sum_1^n X_i$ $U_n=S_n-\lfloor S_n\rfloor$ then $U_n$ is asymptotically uniform in distribution. I've got no idea how to approach this, and it's ...
0
votes
1answer
266 views

Limit of Sum of Cauchy Random Variables

I'm investigating the behaviour of some random variables obtained from standard Cauchy random variables $X_n$. Suppose $Y_n=\textrm{sgn}(X_n)|X_n|^{\alpha}$ for $\alpha\in[0,1]$. Let ...
1
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0answers
57 views

A correction to confidence interval.

I have set of random values with the same distribution $y_1, \ldots, y_N$ , $N = mN_1$. $ m \ge 4$, $N_1$ is big enougth( $\approx 1000$ ). I want to to estimeat $E(x)$. How I do it: I make $m$ ...