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

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30 views

How to fit a gaussian model on the overlapped area of two separate gaussians?

Suppose I have two 1D Gaussians distribution with N(U1,S1) and N(U2,S2) where U is the mean and S is the standard deviation. Suppose if we draw these two Gaussians, they overlap on a interval. Now ...
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0answers
22 views

How to estimate a normal distribution with mixture of gaussians?

I have a set of points which I can fit a Gaussian model on them using Maximum likelihood estimation. but this estimation is weak and I want to improve it. I want to fit a mixture of Gaussian on these ...
1
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1answer
32 views

Standard deviation misunderstanding

Here is a question that I stumbled upon (and the solution to it, my reasoning follows after the image): I answered "The two quantities are equal". My reasoning was as follows: the question mentions ...
1
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1answer
157 views

Bivariate normal distribution of points

I would like to generate points (x,y) in a 2-D plane that has a circular normal distribution similar to this: I found multiple terms for describing a "circular normal distribution" and yet, I'm not ...
1
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0answers
84 views

Normal and poissonian probability problems

I am working on a problem with a normal probability distribution but I am unsure of the results I calculated the probability asked for but still hesitate regarding the output and especially the first ...
1
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0answers
11 views

Finding Conditional Distribution of Multivariate distribution

Q: Supposing y ~ N4 (µ,∑) and µ = (1 2 3 -2)' ∑ =\begin{pmatrix} 4 & 2 & -1 & 2\\ 2&6&3&-2\\ -1&3&5&-4 \\ 2 &-2 &-...
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2answers
402 views

Normal Distribution: Probability of a Negative Value

The random variable $X$ can take negative and positive values. $X$ is distributed normally with mean $3$ and variance $4$. How can I find the probability that $X$ has a negative value?
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0answers
31 views

Proving that a subvector in a multivariate Normal distribution is also a Normal Distribution

Q: Assuming y = [y1 , y2, .. , yp]' is a p-dimensional random vector with mean vector µ and covariance matrix ∑. Given for any a = [a1, a2, .., ap]', we have a'y ~ N[a'µ, a'∑a]. Show ...
1
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1answer
33 views

Random variable with density function that is scaled geometric mean of density functions of two independent normally distributed random variables

Given two independent normally distributed random variables A and B: $$A \sim \mathcal{N(\mu_A, \Sigma_A)}$$$$B \sim \mathcal{N(\mu_B, \Sigma_B)}$$ is there a way to find normally distributed random ...
12
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1answer
104 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad T_N=\sum_{i=1}^{N}X_{i}^{2}\sim\chi^{2}\...
2
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1answer
57 views

Integrating a cost function over a normal distribution

Let's say you have a cost function $C(x)$ and you want to understand the expected cost if the input follows the normal distribution $$X \sim \mathcal{N}(\mu,\sigma ^2)\\ $$ If I want to find my ...
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1answer
21 views

Finding a point on a normal distribution knowing just two other points on it?

So I have two known points on a normal distribution. A 50th percentile point and a 75th percentile point. How can I figure out what percentile a third point is at? For instance: 50th percentile: ...
0
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2answers
110 views

Find the expectation of $Z=min\{X,Y\}$ which $X~N(\mu,{\sigma}^2)$,$Y~N(\mu,{\sigma}^2)$

Find the expectation of $Z=min\{X,Y\}$ which $X\sim N(\mu,{\sigma}^2)$,$Y\sim N(\mu,{\sigma}^2)$. $X$ and $Y$ are independent random variables. This is how far I go: According to order statistics, I ...
1
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1answer
42 views

Finding Marginal Density functions with $Y\sim N_4(\mu,\Sigma)$

Suppose $Y$ is $N_4(\mu, \Sigma)$ where $$\mu = ( 1,2,3,-2)'$$ and $$\Sigma =\begin{bmatrix} 4& 2& -1& 2 \\ 2& 6& 3& -2 \\ -1& 3& 5& -4 \\ 2& -2&...
0
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0answers
38 views

normal distribution under special condition

Given independent Gaussian random variables $U\sim N(−1,1)$ and $V\sim N(1,1)$, are the 2-element vector $T=(U+V, U−2V)$ and the variable $$W= U\text{ with 50% chance}, V \text{ with 50% chance}$$ ...
0
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2answers
52 views

Normal distribution with standard deviation = I

Suppose a vector $\epsilon \in \mathbb R^d$ is a random vector drawn from the isotropic normal distribution: $\epsilon$ ~ $\mathcal N (0, I)$ [As in Eq. 1.34 here.] I suppose ...
0
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2answers
61 views

How does the formula for standard deviation result in the normal distribution

Trying to understand this is in a high school level. I understand that the how $\frac {\Sigma|x-\bar x|}{n}$ calculates the mean of the distances of each score to the mean. I use this idea to map ...
0
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0answers
11 views

How to calculate distribution of (X1, X2) conditional on (C1, C2)?

Say that $X_{1}$ = $a_{1}$$X_{2}$ + $B_{1}$$C_{1}$ + $E_{1}$  , and         $X_{2}$ = $a_{2}$$X_{1}$ + $B_{2}$$C_{2}$ + $E_{2}$  , ...
0
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0answers
71 views

SSE distribution in simple linear regression

I'm looking at the typical simple linear regression model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, where there $\epsilon_i$s are iid $N(0, \sigma^2)$ random variables. We have unbiased estimates $$...
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0answers
46 views

Solving for the limit of a Gaussian random variable within an integral

I'm having trouble solving a particular integral. It is $$ (1/\Delta t)\int_t^{t+\Delta t}I(t')dt', $$ where $$ I(t') = \mu_c+\sigma_c \eta(t'). $$ In this second equation, $$ \eta(t') = \lim_{...
1
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2answers
48 views

Using Normal Distributions to find Proportion

The height of a randomly selected woman from a population is normal with $\mu=165cm$ and $\sigma=7cm$. The heights f the men in this population are normal with $\mu=178cm$ and $\sigma = 8cm$. I am ...
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2answers
26 views

Generating points from 2 Normal distributions and $0$-probability continuous r.v.s

Consider the following experiment: We generate "green" points and "blue" points in $\mathbf{R}$ using two different normal distributions as follows: 1000 green points are sampled from a $N(-1, 1)$ ...
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1answer
443 views

Understanding the matrix normal distribution

A random $n \times p$ matrix $X$ is distributed according to a matrix valued normal distribution iff $\mathrm{vec}(X) \sim \mathcal{N}_{np}(\mu, V \otimes U)$, where $\mu \in \mathbb{R}^{np}$ is a ...
0
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1answer
61 views

If $X$ and $Y$ are Normally distributed with correlation $\rho$, can we say anything about $E[Y \mid X]?$

Let $X \sim N(0, 1)$ and $Y \sim N(0, 1)$ and $\mathbb E[XY]=\rho$. Can one say anything about the conditional expectation $\mathbb E[X \mid Y]$? In general, this clearly does not seem to work, ...
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0answers
132 views

Justify an unbiased estimator is UMVUE

Suppose $X_1,\ldots,X_n$ are iid $N(\theta,\theta)$, with $\theta\in(0,\infty)$. Is $\bar{X}$ the UMVUE (beta unbiased estimator) of $\theta$? I find the complete sufficient statistic is $T=\sum_{i=1}...
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1answer
33 views

expectation of product of sums of normally distr. r.v.

Let $Z_1$ and $Z_2$ be i.i.d. standard normally distributed. $X_1=Z_1+Z_2$ and $X_2=Z_1-Z_2$. Apparantly E[|$X_1|*|X_2|$] = E$[|Z_1|*|Z_2|]$. Why?
0
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1answer
47 views

What is the PDF, CDF, and E[Y] of Y=ln[X+c] if X is lognormal

If $\ln X \sim N(\mu, \sigma^2)$, what is the distribution of $Y=\ln \left(X+c\right)$ where $c$ is a constant. Is this something that can be written out analytically? Also, what is $E[Y]$?
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35 views

Distribution or samples of a function of a random variable

OK I edited the question: I have the following setup: Stereo camera setup with two images I, I'. 4 1-dimensional random variables (each corresponding to the inverse depth value of a pixel on an ...
1
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2answers
52 views

Probability more than 25% greater?

The random variable X is distributed N(60,64). The random variable Y is distributed N(52,36). Find the probability that a random observation from X is more than 25% greater than a random observation ...
1
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2answers
50 views

Estimate mean and variance for a truncated sample set

Assume there is a normally distributed random variable $X \tilde{} N(\mu, \sigma)$ I want to estimate $\mu$ and $\sigma$. So far the standard setting. Assume I am given a sample $(X_i)_{i=1}^N$ of $...
0
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1answer
60 views

What is the correct equation for “Normal distribution function of continuous random variable”?

I was reading a book and came across with a equation which gives the normal distribution function of continuous random variable. It was used in a software called ...
1
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1answer
43 views

T distribution problem

I will be using $t$-distribution to solve this problem. Specifically,the pooled variance test because both samples have size less than $30$,and both populations seem to have the same population ...
1
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0answers
53 views

Integrating a prob distr over the set of possible circles within an annulus

Let $z$ be the measured coordinates of a point on a circle $c$ with center $x$ and radius $r$. Assume the probability of measuring $z$ given the circle $c$ is normally distributed by the distance ...
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2answers
84 views

Covariance and normal distribution

Let $X,Y ∼ N(0,1)$ be i.i.d. and let $U,V$ given by $U=aX+bY+c$ and $V=dX+eY+f$ have a bivariate normal distribution (here $a, b, c, d, e, f ∈ R$ with $ae − bd$ not equals to 0). (a) What is $Cov(X, ...
1
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2answers
36 views

What's $r$ going to be when you get the summation of $36$ Geometric $X_i$'s

Let $X_1,X_2,\ldots,X_{36}$ be a random sample of size $n=36$ from the geometric distribution with the p.d.f: $$f(x) = \left(\frac{1}{4}\right)^{x-1} \left(\frac{3}{4}\right), x = 0,1,2,\ldots$$ Now ...
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1answer
24 views

Definite integral of two dimensional normal distribution

I want to calculate the probability mass in a rectangular area for a two dimensional normal distribution $G(x;\mu,\Sigma)$ . Is it possible to do without numerical integration?
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0answers
79 views

What is the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A company has a one-year group life policy that divides its employees into two classes as follows: Class, Probability of Death, Benefit, Number in Class, A, 0.01, $...
0
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1answer
96 views

Derive a hypothesis test Z

Suppose a random sample of size 10 is taken from the random variable X which has the normal distribution with unknown mean $\mu$ and variance 4. You are requested to test the hypothesis $H_0:\mu = 0$ ...
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0answers
31 views

distribution of infinite sum of independent but non-identical normal variables

For $i=1,2,\ldots,n$, suppose $X_i \sim N(0,\Omega_{i})$, where $\Omega_{i}$ is of dimension $k\times k$. It is known that $\frac{1}{\sqrt{n}} \sum_{i=1}^{n} X_i \sim N(0, \overline{\Omega})$, where ...
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0answers
50 views

partical correlation in mixed case binomial and gaussian

For Gaussian mutlivariate distributions it is known, that zero partial correlation corresponds to conditional independence. Is there a same result if one of the variables has a binomial distribution? ...
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0answers
111 views

$\int_{-\infty}^{+\infty}\phi\left(x\right)\Phi\left(\frac{a}{\mathrm{e}^x}\right)dx=\Phi\left(\frac{a}{\sqrt{2}}\right)$

I think I have found a solution to the integral below using similar logic I have found to an answer here http://mathoverflow.net/questions/101469/integration-of-the-product-of-pdf-cdf-of-normal-...
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2answers
62 views

Normal distribution with sample

I'm trying to figure out the best approach to this problem. I would assume that I can use the Central Limit theorem first and then a binomial cdf: Chocolate is packaged into jars using a computerized ...
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0answers
36 views

Gaussian Bayes Classification with dependent variables..

Gaussian Bayes Classification: two classes: $y \in \{-1,+1\}$ Dependencies for a vector of features ($x_1,x_2,x_3)$: $x_1=z,x_2=2z,x_3=t+3$, where $$P(z\mid y=+1) = \aleph(z;\mu_+,1),\qquad P(z\mid ...
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0answers
11 views

$\exists \mathcal{A},\mathcal{B}:X\sim \mathcal{A}\Rightarrow \frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$?

Does there exist a parametric distribution $\mathcal{A}$, such that: $X\sim \mathcal{A}\Rightarrow\frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$ for some parametric distribution $\mathcal{B}$ Where $p,q,r&...
5
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1answer
111 views

PDF of $X = \max\{X_1,X_2\}$, being $X_1$ and $X_2$ independent Normal distributed random variables

Let $X_1 \sim \mathcal{N}(\mu_1,\sigma_1^2)$ and $X_2 \sim \mathcal{N}(\mu_2,\sigma_2^2)$, what's the CDF of $X = \max\{X_1,X_2\}$? Both variables are assumed to be independent. I tried the following:...
2
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2answers
77 views

If X is log-normal, is: $\frac{a}{\sqrt{b+cX}}$?

I am working for the first time with log-normal distributions and I want to verify whether the following statement is true. I am not sure whether all the properties of the log-normal distribution hold ...
1
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1answer
266 views

Probability - Normal Distribution, Heights of Women versus Men

The heights of young women aged $20$ to $29$ follow approximately the $\mathcal{N}(64, 2.7)$ distribution. Young men the same age have heights distributed as $\mathcal{N}(69.3, 2.8)$. Height is ...
0
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1answer
37 views

Minimum matching convolution (part II)

We assume we are working in $\mathcal{H}(\mathbb{R}^n)$, the space of real symmetric matrices. We define the partial order $\ge$ defined as $\Sigma_1\ge \Sigma_2$ iff $\Sigma_1-\Sigma_2$ is in $\...
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1answer
34 views

Find $P(796.2 \leq \sum_{i=1}^{16}(X_i-50)^2 \leq 2630)$ with a sample of $n=16$ and $X \sim N(50,100)$

If $X_1,X_2, ..., X_{16}$ is a random sample of size $n=16$ from the Normal Distribution $N(50,100)$, determine: $$P(796.2 \leq \sum_{i=1}^{16}(X_i-50)^2 \leq 2630)$$ Okay well I know that $\...
0
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0answers
21 views

how can we generate random numbers using skew normal distribution

I want to generate random numbers with skew normal distribution using rsn(). I can find the answer from the following link. how can we generate random numbers using skew normal distribution in ...