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

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1answer
308 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
114 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
56 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
18 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 ...
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2answers
47 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
37 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
155 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 ...
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66 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
267 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
75 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
72 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
150 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
35 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
66 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
85 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 ...
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3answers
45 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
145 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 ...
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0answers
25 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 $\sigma$ ...
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2answers
83 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 ...
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1answer
83 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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1answer
37 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!!
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1answer
56 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 ...
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2answers
21 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
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1answer
92 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 $$ ...
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1answer
34 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 ...
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3answers
32 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
24 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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1answer
15 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
56 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
19 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) $ ?
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1answer
65 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
22 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 $$ f_2(x) = \frac{1-a}{2 \sigma_2} e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}} $$ ...
0
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1answer
85 views

Likelihood ratio critical region

Let $X_{1},..,X_{n}$ be a random sample from a normal distribution with mean ${\theta}$ and variance 1. We wish to test $H_{0}:{\theta}=0$ vs $H_{1}:{\theta}{\neq}0$. Write down likelihood ratio ...
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0answers
129 views

Limit of sequence of integral related i.i.d. observations

Let $X_1,\dots,X_n$ be i.i.d. random variables, each uniformly distributed on $[0,1]$. Let $\hat F_n$ be their modified empirical distribution function, i.e., $$ \hat ...
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1answer
37 views

Distribution of distance from 0 of gaussian point

Suppose $X_1,...,X_d\sim\mathcal{N}(0,1)$ are i.i.d.'s, each distributed normally around 0 with variation 1. It looks like $\mathbb{E}\left(\sum X_i^2\right)=d$. Why is that true? And how $Y=\sum ...
1
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1answer
221 views

mean and variance normalization of vectors

I have vectors $x \in \mathbb{R}^n$ and I expect some multivariate normal distribution. I want to normalize the vectors in such a way that $y = M (x - b)$ has mean zero ($\operatorname{E}[Y] = 0$) ...
0
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1answer
83 views

Standard deviation with multiple means and deviations

The amounts of a certain mineral that can be produced in a day from mines $1$, $2$, and $3$ are independent normal random variables with means equal to $80$, $90$, and $75$ pounds, respectively, ...
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1answer
25 views

Exponential deviation with two $x$ values

I recently got interested in this topic of standard deviation. My TA did not have any time to go over this topic so I was trying to teach myself it recently. My TA said if he had more time he would ...
3
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2answers
124 views

Integral of exponential using error function

I'm trying to solve some integrals below $$\int_{-\infty}^{\infty} {x^n e^\frac{-(x - \mu)^2}{\sigma^2}}dx$$ I am interested in the solutions where n = 0, 1, 2, 3, 4. I have learned that ...
2
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1answer
292 views

Central Limit Theorem for uncorrelated (non-independent) but bounded random variables

Given uncorrelated, discrete random variables $X_i$ that are bounded, e.g., they can only take on values $|X_i| \leq 4$, then is there a form of the central limit theorem that one can apply to the ...
2
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1answer
52 views

How to find $E[X|X>Y]$

Suppose $X$ and $Y$ are independent standard normal variable. I want to find $E[X|X>Y]$. I calculated that $$ f_{X>Y}(x) = 2\Phi(x)\phi(x)$$ However I couldn't find the expected value using ...
0
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1answer
26 views

Standard normal RV probability

Z is a standard normal random variable I need to find $P(|Z|<.95)$, find c such that $P(|Z|<c)$, and given that X is a RV with mean 3 and standard deviation 16, find $P(X>3.84)$ I am just ...
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0answers
21 views

Finding the distribution of $5X_{1}^2+2X_{1}X_{2}+X_{2}^2$

Suppose $X=[X_{1},X_{2}]$ and $X$~$N_2(μ,Σ)$. I wish to find the distribution of $5X_{1}^2+2X_{1}X_{2}+X_{2}^2$. Since this is of a quadratic form I do not know a way of solving this. However I kind ...
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1answer
48 views

Distribution of log-log linear regression

Edit: Sorry yeah not too clear, probably posted this too late at night... Essentially I have data which appears to be in exponential form - a log-log graph put it close to a straight line. Using R, I ...
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1answer
166 views

Is normalcdf() inclusive?

I was looking at these examples here: Example 1: Given a normal distribution of values for which the mean is 70 and the standard deviation is 4.5. Find: a) the probability that a value is ...
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0answers
54 views

How to simplify the computation of a special case of multivariate normal cdf

I am trying to compute a multivariate normal cdf where all but the last bounds of the integrals are symmetric: $$F(a, \sigma, m ) = ...
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0answers
21 views

Continuity Correction with replacement

An urn contains 2 white and 8 red marbles. A marble is drawn from the urn 100 times in succession with replacement. What is the probability of drawing more than 75 red marbles? My attempt: $n=100, ...
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0answers
35 views

Normal Distribution while finding sigma

I was reading some things about normal distribution and saw this problem in a text a couple days ago. I know it might be a little advanced for me at the moment, but I was wanted to know if someone can ...
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1answer
48 views

If speeds of two cars are Normal RV s, what is the distribution of the distance between them?

The speeds of two cars are random variables that follow $N(\mu_1,\sigma_1)$ and $N(\mu_2,\sigma_2)$ distributions.They start simultaneously. Let X be the distance between them after m hours. (Note ...
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0answers
37 views

How to test a hypothesis which compares set of pairs of statements?

I've conducted an experiment but I'm not sure how to proceed with statistical analysis of it. I have pairs of sentences created by two groups of people A and B, semantically the sentences in each ...