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

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810 views

find probability in normal distribution

i would like to check myself if following my answer is correct: let us consider following problem: Suppose the heights of a population of $3,000$ adult penguins are approximately normally distributed ...
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928 views

calculate probability without table

my question is related to normal distribution,namely as i know in GRE quantity section,there could be question related to normal distribution,but of course we will not have table,o how can we ...
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50 views

Calculating Expectation

I want to verify the following equation: $$E[(xe^{aY-\frac{1}{2}a^2}-b)^+]=x\Phi(l_1)-b\Phi(l_2)$$ where $Y\sim \mathcal{N}(0,1)$, $\Phi$ the distribution function of a standard normal distribution, ...
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108 views

Normal distribution inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. Prove the following inequality. $$(x^2+1)N + xn-(xN+n)^2>N^2$$ where the dependency of $n$ and $N$ on ...
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75 views

Integrals of derivatives of normal distribution multiplied by polynomial?

Is there anything in the literature related to obtaining bounds of integrals of the form: $$\int_{\mathbb{R}} |P^{(k)}(t,z-z_0)|dz\leq \mbox{some function of t and }z_0$$ where $P(t,z)$ the density of ...
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84 views

P.d.f of a discrete fourier transform of binary variables

Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$. The discrete fourier transform is defined $b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
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109 views

Probability that a point from one normal distribution is greater than points taken from several other distributions?

I am looking at several normal distributions that describe the same metric from different sources (independent). I want to find the probability that each is greater than all the others. For example: ...
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123 views

Problem involving the bivariate normal distribution

If $X$ and $Y$ have a bivariate normal distribution with $\mu(x)=\mu(y)=0$, $\rho=0$, $\sigma(x)=\sigma(y)=10$. Find the following: A) The probability of getting a point $(x,y)$ inside the ...
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200 views

Hypothesis testing of normal distribution, known mean unknown variance

I've been working on review problems, and this one has me completely stumped. Let $X_1 ... X_{10}$ be a random sample from a $N(3,\sigma^2)$ distribution, where $\sigma^2$ is unknown. Using the ...
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232 views

Bayesian posterior with integrals over normal densities

Realizations from normal distributions with known precision are used to estimate the mean, but the realizations are not always precisely observed. Instead, only a range of the realization is observed. ...
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36 views

algorithm to use to balance a set of IPs into a set of buckets

So we have a set of IP addresses (~3000) and want to balance them into 4 different buckets. What we are doing now is very simple by treating the last part of the IP as integer and mod it by 4. e.g. ...
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97 views

CAPM-model - necessary conditions for BETA to be a parameter in the conditional expectation

CAPM-model - necessary conditions for BETA to be a parameter in the conditional expectation between the real return on the asset and the stock market return. Okay, trying to be more explicit: Let ...
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78 views

Product of 2 Gaussian Distributions with Different Variables

Sorry, I asked the original question improperly so I am rephrasing it. What is the mean and covariance of the distribution, $f_{PA}(PA) \cdot f_{Y|X,PA}(Y)$ where $f_{PA}$ and $f_{Y|X,PA}$ are both ...
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28 views

What is the product of the average x and the average (1/x), where x is normally distributed.

Has anyone ever seen a solution for the following...? $$ \left( \frac{1}{n}\sum_{i=1}^{n} x_{i} \right) \times \left( \frac{1}{n} \sum_{i=1}^{n} \frac{1}{x_{i}}\right) $$ where the x values are from ...
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240 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 ...
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93 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 ...
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216 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 ...
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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 ...
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169 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 ...
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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 $m$ ...
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167 views

Composition of multi complex gaussian normal distribution

assume $w_0$, $w_1$, $w_2$, $w_3$ are circular symmetric complex Gaussian distributions, and the composite of $$ h = e^{j\theta_0}w_0 + e^{j\theta_3}w_3 - e^{j\theta_1}w_1 -e^{j\theta_2}w_2 $$ so ...
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221 views

Generating spatially correlated samples from a multivariate normal distribution

I am trying to generate some spatially correlated samples from a multivariate normal distribution following this algorithm Compute Cholesky factorization Q=LL' Sample z~N(0,I) Solve L'v=z Computer ...
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521 views

Distribution of a squared norm of related multivariate normal distribution.

For $i=1,2,\cdots,2^m$, let $v_i$ be dependent random variables. Suppose for $n$ large, the vector $\mathbf{Z}_n=\left(Z_1^{(n)},\cdots,Z_{2^m}^{(n)}\right)$ with ...
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233 views

normal random variable distribution

i have such problem in the book of Applied statistic and probability for Enigneering and need some help to solve it.problem is following: Let random variable X denote a measurement from a ...
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118 views

Incrementally compute the conditional entropy

Is it possible to compute a conditional entropy (see the two following formulas) in an incremental manner ? That is, the sets C and K are not fix: each time we have a new element c, set K may increase ...
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126 views

How can I find Multivariate normal distribution original paper? And other nature-revealing articles by the way

I want to see how Gauss get this distribution function representation. I want to understand deeper of Multivariate normal distribution. I tried but failed to search the original paper of Gauss ...
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359 views

Mean value from part of normal distribution

I have a problem to solve. Lets say that there is normal distribution with mean value 5000 and deviation 1000. I have to know lets say what is a mean o 25 percent biggest numbers. How to calculate ...
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253 views

Nested Integral of exponential function with trigonometric identities

Is there any possibility to simplify the following integral or any function that is equivalent to the following integral? $$ ...
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65 views

determine if number belongs in set with multiple normal distributions

I am attempting to work with data with a few of normal distributions mixed in with each other. What I am thinking is splitting up the data in to the separate distributions and then calculating the ...
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670 views

Radial profile from a Cartesian plot

I'd like to calculate a radial profile of a 2D Gaussian. it should be a half of a Gaussian, maximum of about 3000 at $R=0$. If I plot radial positions $\left(\sqrt{x^2+y^2} \right)$ of every ...
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436 views

Approximate linear density function for a normal distribution

I'm working on implementing Order Preserving Encryption for Numeric Data, and part of the algorithm includes approximating density of the distribution as a linear density function $f(p) = qp+r$ where ...
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5 views

Using Convolution to find density of sum of non-independent normal densities

$X_1 \sim N(\mu_1, \sigma_1^2)$ and $X_2 \sim N(\mu_2, \sigma_2^2)$. The $X_i's$ are not independent. Let $Y = X_1$ + $X_2$. Then, $ \begin{align*} f_Y(y) &= \int_{0}^{y} ...
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4 views

Maximum diagonal entry of a multivariate normal sample covariance matrix

Let $\Sigma$ be a covariance matrix of $n$ data points in $\mathbb{R}^p$. So $\Sigma$ is $p\times p$. Suppose that the $n$ points are drawn from the distribution ...
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19 views

How do I set a problem like this up in my calculator?

"It is known that Lemmings (a small rodent like creature) have a mean body weight of 63.5 grams with a standard deviation of 12.2 grams. If the weights are distributed normally find the probability ...
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7 views

How to combine two normal distributions

I want to make a skin detection algorithm based on YCbCr color space. I have a database of $10^7$ triplets (Y,Cb,Cr) which represents a skin color. Now I've computed the normal distribution with ...
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11 views

Applying homography to ellipse derived from normal distribution

I need to apply a homography to an elliptic area. First question: Is the resulting also elliptic in every case? I think so, but actually i don't really know. Anyway, I assume it for this question. ...
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47 views

Normal distribution in nature

I applied for a job as a mathematician. In one of the test questions they asked the following: Why normal distribution is so common in nature? What do you think?
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7 views

Draw an ellipse corresponding to a bivariate normal distribution

Let $$\mu=\left(\begin{array}{c}\mu_1 \\ \mu_2\end{array}\right)$$ and $$\Sigma=\left(\begin{array}{cc}\Sigma_{1,1} & \Sigma_{1,2} \\ \Sigma_{2,1} & \Sigma_{2,2}\end{array}\right)$$ be ...
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0answers
24 views

Estimate prior for normal distributed data

I have successfully implemented a bayesian classifier using maximum likelihood. In my case I've got 2 classes and I have calculated the two $\mu$ and $\Sigma$. In my problem with a 3-dimensional ...
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0answers
25 views

Normal Distribution of a random variable

Given that a random variable is distributed normally with E(X)=-1 and p(-7<=X<=-2)+p(1<=X<=3)=0.33. Find p(-7<=X<=-2). Please assist me with the steps in solving this problem and ...
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12 views

Estimate number of data points necessary to generate a normal distribution

I've written a program that generates random normally-distributed variables using the Box-Muller transform. My question is if I can find any formula that relates the number of data points that I ...
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18 views

Attempted calculation of the probability to win a game.

I'm playing the game "Pepper Panic" and the goal is to create two pepper panics. I noted down some ten results by the numbers I obtained ($0$ or $1$). I obtained a mean of $0.4$ and a standard ...
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7 views

Subset of samples has any effect on sufficiency of the statistic?

If we have the following iid samples $$ X_1, ..., X_n \sim N(\mu, \sigma^2) $$ where only $\mu$ is unknown. We know one sufficient statistic is the following: $$ T = \frac{1}{n} \sum_{i=1}^n X_i $$ ...
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18 views

convergence of sequence of functions with finite second moment

Given $0<a<1$. Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$. Suppose we are given a sequence of functions $\{f_n\}$ ...
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0answers
7 views

How does one scale a covariance matrix learned on de-meaned and scaled data?

I have a dataset on which I want to train a multivariate mixture of gaussians. One common thing to do is de-mean and scale the data such that each feature has zero mean and unit covariance. If I ...
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0answers
11 views

Combining Two Gaussian Filters

I am taking a class related to image processing and we were taught about Gaussian Filters that are related to the following Gaussian Function: $$G(u,v) = \frac{1}{2\pi\sigma^2}e^{-\frac{u^2 + ...
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36 views

10 random observations

I have to generate $10$ random numbers with mean $= 22.6$ and standard deviation $= 2.4$. Then I have to see if the $10$ observations appear to be normally distributed. When I generate $10$ random ...
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18 views

Certain parasite Trypanosome

Consider the length of individual trypanosome chosen at random from the population. Find a) $\def\Pr{\operatorname{Pr}}\Pr\{20 < length < 30\}$ I just say 0.41 + 0.21 = 0.62 b) $\Pr\{length ...
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9 views

Normal distribution around and extreme value

I'm trying to create an artificial dataset with users, items, and ratings given by the users to the items. Creating the dataset, I pick the average rating for every item randomly, and let the ratings ...
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18 views

What type of distribution can be used to describe a game with positive expected winnings?

I've come across something I'm not too sure about. Let's say we flip a coin, heads mean we lose 1 unit, tails means we win a 1 unit. This distribution of outcomes in this would be considered normal, ...