Questions tagged [normal-distribution]

This tag is for questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution. The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.

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How can I make a values out of two groups that have seperate normal distributions?

I am doing a graduation research and got stuck for two weeks solving this problem. I am not a expert in mathematics so I might use the wrong expressions. I have sensor data that measures the movement ...
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computing the expected value of two dependent normally distributed variables

Let us assume that $D$ ~ $N(\bar{D}, \sigma^2)$ and $z$ ~$N(0, \sigma_z^2)$ D and z are dependent variables How can calculate $E(D z)$ = ? Thank you!
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About explicitly constructing a probability space and a random variable which has normal distribution

I am reading a book on Brownian motion. It assumes, for the proof of the existence of the Brownian motion, a probability space $(\Omega, \mathcal{A}, P)$ exists on which we can define a collection $\{...
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Complete the square of $-n\theta+2\theta\sum{ln(x_i)}$ and then substituting for normal distribution

I am revising old exercises where the teacher has completed the square of what's inside the $\exp[]$ expression. $p(\theta|x) \propto \exp(-n\theta+2\theta\sum{ln(x_i)})$ in his solution, but not ...
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Given two IID N(0,1) variables X and Y, what's $P(X+2Y \geq 0 | X \geq 0)$?

My attempt is that $$ \begin{aligned} P(X+2Y \geq 0 \mid X \geq 0) &= P(Y \geq 0) + P(Y\leq0 \text{ and }|Y| < X/2 \mid X \geq 0)\\ &= 1/2 + P(-X/2 <Y < 0 \mid X \geq 0)\\ &= 1/2 ...
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Fast approximation for bivariate normal cumulative distribution function

The cdf of a normal distribution has some nice approximations like $$F(x) = 1/(1 + \exp(-0.07056 x^3 – 1.5976 x)),$$ see https://www.econstor.eu/bitstream/10419/188388/1/v02-i01-p114_60-313-1-PB.pdf ...
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Why are Some Probability Distributions "Used More Often" for Certain Things?

I have had this question for a long time - Why are Some Probability Distributions "Used More Often" for Certain Things? I understand that for certain problems, the type of probability ...
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Scaling of moments of normal distribution/Brownian incremenrs

Let $B_t$ be Brownian motion for $t\ge0$ so that $B_t-B_s \sim N(0,t-s)$. There are a few formulae out there that show how the $p$-th moment of $E[(B_t-B_s)^p]$ is calculated. My question is: how do ...
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Mutual Information between $v_1$ and $v_2$ coming from the same Inverse-Wishart distribution?

Say that $\left(\begin{matrix} v_1 & c\\ c & v_2 \end{matrix}\right)$ is a bivariate covariance matrix that comes from an Inverse-Wishart distribution $W^{-1}(\Psi, \nu)$. Then what is the ...
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Understanding absolute value of random variables [closed]

In my work I am facing the following situation: $y = |a+bc|^2$ ----(1) where $a,b,c$ are zero mean circularly symmetric complex Gaussian (ZMCSCG) random variables with variance $\sigma^2_a, \sigma^2_b,...
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PDF of following random variable.

I am trying to find the PDF of random variable $X$ but not getting it correctly. $X = \sum_{m=1}^{N}|a_m+\eta b_m c_m|^2$ ----(1) where $a, b, c$ are Zero mean circularly symmetric complex Gaussian (...
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Find estimates for $\alpha_0$ and $\alpha_1$ and covariancematrix

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $$f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$$ and we ...
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What is the distribution of a quadratic function of normal distributions? [closed]

Suppose $Z_i$ is independent standard normal distributions, i.e. $Z_i\sim N(0,1)$, $i=1,2,\cdots, d$. What is the distribution of $$ \sum_{i=1}^d (a_iZ_i+b_iZ_i^2). $$ I know when $a_i=0$, it is the ...
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Find asymptotic variance of moment estimator

I have that $$f(x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}x^2}$$ I have the conditional distribution: $$f_{\beta}(y|x)=\frac{1}{\sqrt{2 \pi}}e^{-\frac{1}{2}(y-\beta_0-\beta_1x-\beta_2x^2)^2}$$ and we ...
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Should I reject the null hypothesis or not?

EDIT: My apologies, I had a coding error. I accidentally used the same standard deviation for both samples. Now that I fixed that, both the normal and Student's confidence intervals are stupidly ...
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Multivariate Gaussian Derivation [closed]

I am learning multivariate Gaussian as my interest during leisure time. I don't understand how to go from the definition, which says a random variables X belongs to R^n is a multivariate Gaussian if ...
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Distribution of $Z = \frac{\sin (XY)}{\cos (X+Y)}$ where $X$ and $Y$ have a joint bivariate normal distribution?

Let us suppose that $X$ and $Y$ have a joint bivariate normal distribution with mean vector $\vec{\mu}$ and covariance matrix $\Sigma$. What distribution does $Z$ have if $$Z := \frac{\sin (XY)}{\cos (...
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Variance of product of Gaussian random variables

Suppose I have $r = [r_1, r_2, ..., r_n]$, which are iid and follow normal distribution of $N(\mu, \sigma^2)$, then I have weight vector of $h = [h_1, h_2, ...,h_n]$, which iid followed $N(0, \...
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2 votes
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$exp$ of Half-Normal Distribution

I know that the Half-Normal Distribution has moments of all orders - that is, if $X\sim\mathcal{N}(\mu,\sigma)$, then, $$ E[|X|^p]<\infty $$ However, do we also have $$ E[e^{|X|}]<\infty $$ ? ...
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Weaker version of Central Limit Theorem

The Central Limit Theorem applies to i.i.d random variables. I’ve seen mention of a version of the CLT that applies to the sum of random variables that while independent, are not identically ...
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Bound Hypergeometric distribution by Gauss distribution

I want to lower bound the pdf of a Hypergeometric distribution $H(N,K,K)$, which has equal number of success states and number of draws with the pdf of a normal distribution. With the central limit ...
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2 votes
1 answer
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Expected value of maximum of two numbers, where one is normal distributed

I am searching something similar to the first order loss function ( $\mathbb{E}[max({y_{i}-y_{i},0})])$ (where $y_{i}$ is normally distributed) but for $\mathbb{E}[min(y_{i},d_{i})]$ , where $y_{i}$ ...
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Stochastic Ordering of multivariate normal distribution

Let $$X\sim\mathbb{N}([0,1,0]^{\rm T}, \mathrm{\Sigma})$$ and $$Y\sim\mathbb{N}([1,0,0]^{\rm T}, \mathrm{\Sigma}).$$ For some real constant $c$ what can be said about $$\mathbb{P}[\cap_{i=1}^3|X_i|\...
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4 votes
1 answer
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Why $[(\mathbf{I}_N-\mathbf{A}^\top \mathbf{A})\mathbf{x}]$ is Gaussian with i.i.d. Gaussian $\mathbf{A}$?

1. Background: It is presented in the paper of approximate message passing (AMP) algorithm [Paper Link] that (the conclusion below is slightly modified without changing its original meaning): Given a ...
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$L^2$ Approximation error in Gaussian Process Regression (finite data setting)

I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...
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How do I calculate the probability with standard deviation?

I dont quite understand how to solve this task? Can anybody be of help?
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why do I have to use normal distribution in this exercise?

on average, a biker inflates their bike tires every 8 days. the interval between each time, namely t1, t2, and so on, is an exponential random variable. Find probability that 40 inflations are ...
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1 vote
1 answer
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Changing limits wen integrating with exponential of normally distributed variable

I am working on an exercise which requires calculating the expected value of : $ E[e^{\mu + \sigma Z} \mathbb{1}_{Z > -d}] $ Calculating the expected value of the $e^{\mu + \sigma Z}$ variable and ...
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1 vote
1 answer
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Almost sure convergence of a sequence of Gaussians with variance sequence

I want to prove the following: Consider a sequence of $(Z_n)_{n \in \mathbb{N}}$ of independent random variables such that $Z_n \sim \mathcal{N}(0,\sigma_n^2)$. Now let $S_n = \sum_{j=1}^n Z_i$ and $\...
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4 votes
1 answer
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Let $X,Y,Z$ be three random variables such that the correlation coefficients $\rho_{XY}=0.2, \rho_{YZ}=0.2$, what values can $\rho_{XZ}$ take?

Let $(X,Y,Z)^T$ be jointly normal variable with zero mean such that the correlation coefficients $\rho_{XY}=0.2, \rho_{YZ}=0.2$, what values can $\rho_{XZ}$ take? Prove that there exists a ...
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Combining non-independent random normal-distributed variables

In the below question part b) involves combining normally-distributed random variables which ARE independent. Part d) involves combining normally-distributed random variables which are NOT independent....
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1 vote
1 answer
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calculation of $E[\Phi(X)]$

Let, $X\sim N(\mu, \sigma^2)$ and let $\Phi(\cdot):\mathbb{R}\to[0,1]$ be the CDF of a standard normal distribution. Then, what is the pdf of $Y=\Phi(X)$. Also, find $E[\Phi(X)]$. Note:- Here, $Y=\...
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How do I prove that the distribution of $x_t$ in $x_t=x_{t-1}e^{-μt}+ θ(1-e^{-μΔt}) + \int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ is a normal distribution?

$B_s$ is brownian motion. Because $\int_{t-1}^t e^{-μ(t-s)}\sigma dB_s $ has a Brownian component, it is normally distributed according to Taylor & Karin 1998 's Introduction to Stochastic ...
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Normal Distribution of 2 mutually exclusive random variables

Can someone give a hint or explanation on how to approach this question? Let $X_1$ and $X_2$ be two independent random variables that each of them is normally distributed $N(6,1)$ and $N(7,1)$ ...
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Coefficients of posterior on a prior which is gaussians mixture

Having known about sample $x \sim \mathcal{N}(\theta, 1)$ and $\bar{x} = -0.5$, size $n = 50$, a prior $p(\theta) = 0.95\mathcal{N}(1, 0.5^2) + 0.05 \mathcal{N}(-1, 0.5^2)$, I need to show that $p(\...
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Is there a such thing as "Stacked Vector Notation"?

I was reading this link over here: https://peterroelants.github.io/posts/gaussian-process-tutorial/ . I came across the following statements: I am trying to understand how to "fill in the ...
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"Manipulating" Normal Distributions

I am reading the following book https://algorithmsbook.com/optimization/files/optimization.pdf at page 281: I am trying to understand how to manipulate the matrix terms to verify the following 2 ...
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2 votes
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Relationship Between Bayesian Optimization and Gaussian Process

In Bayesian Optimization, the function (i.e. objective function) that we are trying to optimize is modelled using some surrogate function - this surrogate function usually turns out to be a Gaussian ...
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Proving Lemmas in Gaussian Distribution

I am struggling to prove the following lemmas: How would you suggest me to solve it?
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Inverse Normal Distribution, why is this z value negative?

I have a Normal Distribution with unknown mean: D ~ N(u - 30, 28.09) and I know P(D < 4) = 0.1 If I use the Inverse Normal Distribution with u = 0, s.d. = 1 and area = 0.1 I get z = 1.2816 But in ...
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Normal distribution with a conditional probability

Given this Normal Distribution: Find: RE the conditional probability I don't understand how the mark scheme got the numerator. If we have P(120 < B < 146) then we get: = P(B < 146) - P(B &...
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Confused with following statements

While reading this statistics statement, I am pretty confused. When performing a test of significance about a population mean, a t-distribution, instead of a normal distribution, is often utilized. ...
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Conditional Expectation of Squared Normal Tail

Let $X$ be a normally distributed random variable with mean 0 and variance $1$. Let $\lambda > 0$. What is the value of the conditional expectation $$ E[X^2 | |X|>\lambda]?$$ I got the following ...
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Simplifying a Kullback-Leibler divergence

I want to find an approximation of a mixture of probability distributions that minimises the Kullback-Leibler divergence (KLD). I need to verify my result, as it seems suspect. We have a joint ...
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Integral of the normal distribution multiplied with a density function

I am trying to solve an integral. This integral is pretty difficult to solve. I tried it with the help of an integral calculator, but the calculator shows the solution with the error function.... I ...
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Random sample from Normal Distribution, why isn't the variance divided by n?

Below is a question containing parts b) and c). Further below is the mark scheme. In part c) they take a random sample from T, which is normally distributed. I was expecting the distribution to change,...
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1 vote
1 answer
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Normal Distribution, negative Z, trying to find variance

I have a D ~ N(5, 27 + sigma^2) and P(D < 0) = 0.2. I need to find the variance. I get this far: However, I am stuck how to convert to an equivalent formula containing only Z values. The answer ...
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A basic inequality of normal pdf and cdf

$\phi(\cdot)$ and $\Phi(\cdot)$ are the pdf and cdf of the standard normal variable, respectively. Then $\forall a<0$, is it true that $\Phi\left(a\right) \leq \phi\left(a\right)$?
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How to derive the von Mises-Fisher distribution by restricting an isotropic normal distribution to a the unit sphere?

I've heard that one way of characterizing the von Mises-Fisher distribution is to restrict an isotropic normal distribution to a unit sphere. How is this done in practice? I know that the density ...
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3 votes
1 answer
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Exponential bound for tail of standard normal distributed random variable

Let $X\sim N(0,1)$ and $a\geq 0$. I have to show that $$\mathbb{P}(X\geq a)\leq\frac{\exp(\frac{-a^2}{2})}{1+a}$$ I have no problem showing that $\mathbb{P}(X\geq a)\leq \frac{\exp(\frac{-a^2}{2})}{a\...
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