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

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1answer
25 views

Moment Generating Function of Gaussian Distribution

Derive from first principles, the moment generating function of a Gaussian Distribution with $$PDF= \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{-(x- \mu)^2/2\sigma^2}$$ MY ATTEMPT MGF= E[$e^{tx}$]= ...
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0answers
25 views

Maximisation of Conditional Gaussian Mixture Model using EM Algorithm

Assume, the pdf of conditional Gaussian mixture distribution of $X_{A}$ given $X_{B}$ is formulated as follows: $f(X_{A}/X_{B}) =\sum^{K}_{k=1} ...
0
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0answers
21 views

For which joint distributions is a conditional expectation an additive function?

I know that, for a random vector $(X,Y,Z)$ jointly normally distributed, the conditional expectation $\mathbb{E}[\,X\mid Y=y,Z=z]$ is an additive function of $y$ and $z$. For what other distributions ...
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2answers
75 views

Distribution of mean of Normal distribution

Suppose $X\sim N(\mu,\sigma)$. I want to find the following probability $P[\mu \ge \theta |x= \theta -c]$ for $c>0$. In another word, I saw a sample of Normal distribution, $x$, and know that it ...
0
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1answer
18 views

Does correlation have to be in the context of (Gaussian) normal distribution?

I am not quite familiar with the concept of correlation. The Pearson's correlation coefficient is defined as: $\rho_{X,Y}=\mathrm{corr}(X,Y)={\mathrm{cov}(X,Y) \over \sigma_X \sigma_Y} ...
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0answers
47 views

Expectation of maximum of product of normal random variables

Let $X_i Y_i \sim N(0,\sigma^2) N(\mu,\sigma^2b)$, $\mu \neq 0 ,b >0$. then is there any inequality for the maximum of these products. What I mean is $E(\max{X_iY_i, 1 \leq i \leq m})$. I have ...
0
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1answer
15 views

Show that $Z-\tilde{Z}_{\iota_{\nu}}$ and $\tilde{Z}$ are independent.

Let $Z\sim N(a\iota_{\nu},I_{\nu}), a\in\mathbb{R}$ whereat $$ \iota_{\nu}=\begin{pmatrix}1\\1\\\vdots\\1\end{pmatrix},~~~I_{\nu}=\text{diag}(1,\ldots,1). $$ Show that ...
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0answers
37 views

Bivariate normal distribution; rotation; diagonal covariance matrix

Let $Z\sim N(0,\Sigma)$ with $$ \Sigma=\begin{pmatrix}\sigma_1^2 & p\sigma_1\sigma_2\\p\sigma_1\sigma_2 & \sigma_2^2\end{pmatrix} $$ whereat $\sigma_i^2=\text{var}(Z_i), ...
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0answers
13 views

Controlling a random variable

I've got a system the output of which is a random variable with a certain distribution, which for the purpose of this discussion can be assumed to be normal. The input variable is voltage. The ...
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0answers
15 views

Binomial distribution vs Normal distribution

It is often said that the normal distribution "approximates" the binomial distribution. What is the precise mathematical expression of this fact?
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0answers
27 views

Covariance matrix and Gaussian i.i.d. random variables

I have a set $X = \left \{ X_i | i \in (1,n) \wedge X_i \text{ is a random variable} \right \} $ Does $\forall i \in (1,n ), X_i \text{ follows a normal distribution} $ implies that ...
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0answers
29 views

How to compute the expected value of normal distribution over a finite interval.

The occurrence time of event A is normally distributed with mean $\mu=200$ and variance $\sigma^2=10^2$. That is, $f(A) \sim \mathcal{N}(200, 10^2)$. As known, the expected occurrence time of A can ...
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0answers
12 views

Expectation of product of Normal CDFs w.r.t. a bivariate Normal distribution?

I am trying to figure out if there is a closed form expression for the following expectation: $\int\int \phi(\gamma_1)\phi(\gamma_2) \mathcal{N}(\gamma\big|\mu, \Sigma)d\gamma_1 d\gamma_2$ where ...
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1answer
36 views

Finding probability density function of a linear combination of mutually independent normal random variables

I'm finding the probability density function of the random variable U defined in the following manner: $$U=\frac{1}{2}(Y_1+3Y_2)$$ CORRECTION: The line above is supposed to be ...
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0answers
46 views

Expectation of normal CDF with truncation

Suppose that $a$ and $T$ are given positive numbers. I would like to evaluate $$\begin{align*} \mathbb{E}\left[\Phi\left(aX\right)\mu\left(X+T\right) \right],\tag{1} \end{align*}$$ where ...
1
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1answer
31 views

Normal distribution without standard deviation given

The proportion of pink candies in a bag is supposed to be $50\%$. The filling machine is to be tested to see if it fills with the right proportion. A random sample of $50$ candies is made. The machine ...
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0answers
14 views

Modeling Gaussian Error

Context I am designing a simulation of a robot receiving input from a sensor which has gaussian error. The robot will start from a known position and move at a constant speed; the sensor will ...
1
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2answers
39 views

When do normal distributions not occur?

I know that in many cases one can assume a normal distributed probability density. But what the situations when the distribution in non-normal. Some examples would be nice. For example, suppose ...
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0answers
18 views

Estimation for expected value of a combination of two normal distributed random variables

I am struggeling to understand the proof of the following Lemma. Let $\epsilon_1$, $\epsilon_2$ be $\mathcal{N}(0,1)$ random variables. Then $\forall$ b $\geq$ 0 and $\forall$ c $\geq$ 1 there is an ...
4
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1answer
129 views

Probability of a gaussian distribution in another gaussian distribution

Assume we have a Gaussian distribution $p(x) \sim \mathcal{N}(\mu_p,\Sigma_p)$ For any point $X$, it is easy to compute the density of $x$ in $p$: $$p(x) = \frac{1}{|2\pi ...
3
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0answers
60 views

Independent normal distributions

I found two theorems with a similar content and want to find out which one is true: Let $X,Y$ be normally distributed random variables and $X+Y$ is also normally distributed or $ (X,Y)$ is ...
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0answers
37 views

How to use the normal probability table in reverse

I'm just wondering if anyone could give me a bit of advice on this. This relates to CCEA's S1 exam questions. $Z \sim \text{N}(0, 1)$ Let's say $\phi(z) = 0.5015$ Find z. Here is an extract of the ...
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1answer
36 views

moment generating function of normal distribution

I know this question relates to the chi-squared distribution, but I think what the question wants me to do is somehow derive this distribution from the information given. I have a normally ...
2
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1answer
63 views

Showing that a certain stochastic process does not have normal distributed increments

Edit: Question Resolved. See below. As a part of my bachelor thesis, I have to work through a paper about fake Brownian motion by Oleszkiewicz. In this paper he defines a stochastic process. Let ...
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2answers
48 views

Bivariate normal distribution question

If I have $(X,Y)$ with joint density $f(x,y)$ and $A$ is an invertible $2\times 2$ matrix, then for the random vector $(W,V)$ defined by: $$ \begin{pmatrix} W\\ V \\ ...
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2answers
29 views

Defining the domain of an MGF?

Let $Y=X^2$ and let $X$ follow a distribution of $X\sim N(0,\sigma^2)$ for $\sigma > 0$. Find the MGF of $Y$ and specify its domain. So what I did was I did a change of variables: ...
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0answers
30 views

Integral of multivariate normal density function

Is anybody know a suited close-form solution for this integral: $$ I=\int_{R^n} x_i \cdot x_j \cdot f_N({\bf x},{\bf \mu},{\bf \Sigma}) d{\bf x} $$ where ${\bf x}=\{x_1,\ldots,x_n\}$ and $f_N$ is the ...
2
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1answer
47 views

Calculation of distribution of a gaussian process

Currently finishing the last year of PhD in statistics, we wonder if you could help us with the following. Let $T = [0,1]$ and $X = \left( X_{t}, t \in T \right)$ be a gaussian process with mean ...
-1
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1answer
52 views

What's the pdf of $Z=X^2 +2X$ if $X$ is a standard normal? [closed]

Le be $X$ distributed as a standard normal. What is the density function of $Z=X^2 +2X$? Thanks for your help
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3answers
37 views

Independent variables, normal distribution, pdf

I have independent variables $ X_1, X_2,\ldots,X_n $ with normal distribution on range $ [0,1] $ . Next, variables $ Z_i $ are created according to this formula $ Z_i = - \frac{1}{\lambda} \ln(1-X_i) ...
5
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1answer
94 views

Estimating a gaussian distribution from a GMM

Suppose that we have a Gaussian mixture model (GMM) in n-dimensional space: $$P_1(x) = \sum_{i=1}^{C}\pi(c_i)\mathcal{N}(\mu_i,\Sigma_i)$$ We want to estimate a single Gaussian distribution from ...
0
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0answers
12 views

Proving limit of variance estimator (Normal distribution)

i have a problem with a exercise from my statistics I book, some help would be appretiated ... Let $x_1,x_2,...,x_n$ random sample from normal distribution $N(\mu,\sigma)$, where $\mu$ and $\sigma$ ...
0
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2answers
29 views

Numerical precision of product of probabilities (normal CDF)

I'm trying to calculate $\prod_k{p_k}$ where $p_k$ are (potentially) very high probabilities of independent, zero-mean, standard normal random variables and $k>100$. However, I'm running into ...
0
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1answer
27 views

P-P plot and Q-Q plot

How to draw P-P plot and Q-Q plot manually ? I have looked at different site and they explained in various way, such as one said for p-p plot in X-axis there is residual in ascending order and in ...
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0answers
7 views

Upper 5% value of distribution of the mean

The density function of a random variable x is $f(x)=ke^{-2x^{2}+10x}$. Find the upper 5% point of the distribution of the means of the random sample of size 25 from the above population. I need ...
1
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1answer
21 views

limiting behavior of standard normal survivor function [duplicate]

How do you show that $\lim_{x\to \infty} 1-\Phi(x) \sim \phi(x)/x$? In the previous, I'm using $\Phi$ to refer to the standard normal CDF and $\phi$ to refer to the standard normal pdf. Thanks!!
0
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1answer
39 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 ...
0
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2answers
14 views

Determining the marginal distribution

Consider $X=(X_1,\ldots,X_n)^T\sim\mathcal{N}(\mu,V)$. Show that then $X_i\sim\mathcal{N}(\mu_i,V_{ii})$ for all $1\leqslant i\leqslant n$. Good day! Ok, I have to determine the marginal ...
2
votes
1answer
60 views

Upper bound of difference of squares of quantile standard normal

Let $\Phi$ denotes the cummulative standard normal distribution and $\Phi^{-1}$ denotes its inverse. Given $u,v\in[0,1)$. I'am going to find an upper bound of $$ ...
1
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1answer
30 views

Probability distritubion of linear function

Given a variable X belongs to gaussian distribution $N(\mu, \sigma)$. How to find the distribution of linear function $y=ax+b$? My answer is that the linear distribtion will belong the ...
0
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0answers
14 views

Show that $E(Z)=\mu, Cov(Z)=V$

Assume that $Z=(Z_1,\ldots,Z_n)^T\sim N(\mu,V)$. Show that $E(Z)=\mu$ and $Cov(Z)=V$. Is assumed that $Z_i\sim N(\mu_i,\sigma_i^2)$? And that the $Z_i$ are not corolated? Then ...
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0answers
11 views

Show that $Z_i\sim N(\mu_i,V_{ii})$

Consider $Z=(Z_1,...,Z_n)^T\sim N(\mu,V)$ Show that $Z_i\sim N(\mu_i,V_{ii})$ for any $i=1,...,n$. I know I have to calculate $$ f(Z_i)=\int\ldots\int ...
1
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3answers
28 views

Show that $Y\sim N(a+A\mu,AVA^T)$

Consider $Z=(Z_1,\ldots,Z_n)^T\sim N(\mu,V)$. Show: If $a\in\mathbb{R}^m$ and $A$ is a $(m\times n)$-matrix with $\text{rang}(A)=m$ then $$ Y=a+AZ\sim N(a+A\mu,AVA^T). $$ My ...
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0answers
11 views

Properties of the multivariate normal distribution

Consider $$Z=(Z_1,\ldots,Z_n)^T\sim\mathcal{N}(\mu,V).$$ 1. Show that $\mathbb{E}(Z)=\mu$ and $\mathbb{Cov}(Z)=V$. 2. Show that for $d\in\mathbb{R}^n$ it is ...
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0answers
8 views

Uniqueness of zero point for a function related to normal cdf

Let $a,b,c$ be three given real numbers with $0<a<b<c$. $\Phi$ is the cdf for standard normal distribution. Let $A_1 = \Phi (x+a) - \Phi (x)$, $A_2 = \Phi (x+c) - \Phi (x+b)$, and $\delta(x) ...
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1answer
12 views

Normally distributed data or not

Can I say that the datas are normally distributed? I would say yes, but I am not entirely sure.
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2answers
35 views

Binomial and Normal Distribution Problem - Check solution

Whooping cough is a highly contagious bacterial infection...About 80% of unvaccinated children who are exposed to whooping cough will develop the infection, as opposed to only about 5% of vaccinated ...
1
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1answer
14 views

Normal Distribution how $N(x-x_n|0,\sigma^2) = N(x |x_n,\sigma^2) $

I read an expression Could someone explain the step $N(t-t_n|0,\sigma^2) = N(t | t_n,\sigma^2) $ ?
1
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1answer
24 views

A practical question in statistics

A student leaves home at 8 a.m. every morning in order to arrive at the University at 9 a.m. He finds that over a long period he is late once in forty times. ($\frac{1}{40}$) He then tries leaving ...
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0answers
16 views

What is the minimum standard deviation for a normal PDF such that one tail is always larger than that of a second normal PDF (different means)?

Say I have two weighted normal distributions, $$ f_1(x) = \frac{a}{2 \sigma_1} e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}} $$ and $$ f_2(x) = \frac{1-a}{2 \sigma_2} e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}} $$ ...