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Questions tagged [normal-distribution]

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

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Special expression for solution of heat equation

Let us consider the heat equation on the quarter plane \begin{align*} \frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2},\quad t>0,\;x>0 \end{align*} with boundary condition $u(0,t) ...
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Proving expectation of Centered Gaussian RV conditional on positive Gaussian RVs with positive covariances is positive

$Y \sim N(0, \Sigma)$, for $Y \in \mathbb{R}^D$ with $\mathrm{cov}(Y_i, Y_j) > 0$ for all $i, j \leq D$. I wish to prove that $$\mathbb{E}[Y_1 | Y_2 > 0, \dots, Y_D > 0] > 0.$$ Its seems ...
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Complicated Demonstration - Violation of the theorem that converges in probability and not in distribution

I was thinking that if a sequence of random variables $Y_n$ with c.d.f. $H_n$ which converges to $c$ in probability, such that $H_n(c)$ does not converge to $H(c)=1$. How could I make an example ...
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Goodness of fit test

I have the following exercise that shows $n=6$ numbers: $$ 1.40, 1.55, 1.35, 1.50, 1.29, 1.64 $$ Is data normally distributed at the 5% significance level? Surely $\overline{x} = 1.455$, $s=0....
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Difficult demonstration - How to show that $H_n$ is normal distributed $N(\xi,\sigma^2)$ starting from its moments $ξ$ and $σ$?

I was thinking that if the function $H_n$ of cumulative distribution converges to a distribution $H$, then $\epsilon_n$ should converge to $\epsilon$ what could be expressed as follows: If $H_n$ is ...
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Finding the percentage of bolts likely to be defective

2000 bolts have a mean width of 10mm and a standard deviation of 0.2mm, a bolt is defective if it is less than 9.5mm. Find the percentage of bolts that are likely to be defective. So far I have; z = ...
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Convergence (to zero) for PDF of normal distribution.

I need to prove that the PDF converges to zero when $n\to\infty$; that is, $$\lim_{n\to \infty}f_n(x) =\lim_{n\to\infty} \frac{1}{\sqrt{2\pi n^{-3}}}\exp\left(-\frac{(x-\frac{1}{n})^2}{2n^{-3}}\right)...
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Geometric distribution, can this approach be correct?

I am revising and came upon these questions, but I am not sure if the answers are right especially for part B Suppose the probability of defective is p(D) = 0.009 , p(d^c)=0.991 and mean 7 and std = ...
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1answer
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How do you find the probability that the sample mean is between 52 and 56?

I was attempting to help a friend with a question and I am not sure if I am overthinking it, or simply missing an assumption I can make. It goes like this: The average life of a battery is 50 hours ...
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1answer
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Estimating mean from a biased sample

Imagine that somebody had chosen $N$ numbers from a normal distribution with mean $\mu$ and variance $1$ ($\mu$ is unknown to you) and only showed you all $n \le N$ numbers which are greater that $\mu$...
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Find $k \in R$ such that $P\left(\max\left\{\frac{{S_x}^2}{{S_y}^2}, \frac{{S_y}^2}{{S_x}^2}\right\} > k\right)= 0.05$

Let $\overline{X}$ and $\overline{Y}$ sample means and ${S_x}^2, {S_y}^2$ unbiased estimators for the variance of 2 independent random samples of size 7 with normal distribution with mean unknown and ...
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Visual Intuition: Gaussians closed under addition

I'm trying to develop some intuition for the fact that the family of Gaussian distributions is closed under addition. I.e. if $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$, then $Y = \sum_iX_i$ is also ...
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Doubt about distribution of the brownian motion

Let $B_{t}$ a brownian motion (stochastic process) then I know $B_{t} -B_{s}$ has a normal distribution with mean$=0$ and variance $=t-s$ I want to calculate the following probability: $P(3B_{2}>4)...
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If $\Sigma^{-1}=(A^{-1})^TA^{-1}$, then why does $|A^{-1}|=|\Sigma|^{-1/2}$?

In the derivation of the joint pdf of $f_\textbf{X}(\pmb{x})$, where $\textbf{X}=\pmb\mu+A\pmb Z$ and $\textbf{X}\sim~N_n(\pmb\mu,\Sigma)$, there is a step I do not understand. In particular, it is ...
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preservation of stochastic dominance by convolution with normal distribution

Let $\mu , \nu$ be probability measures on $\Bbb R$ with symmetric densities with respect to the lebesgue measure. Let us assume that $\mu ([-c, c]) \geq \nu ([-c,c])\ \forall c \geq 0$ (So $\mu$ ...
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Finding Standard Deviation in Normal Distribution [closed]

Please help, i've just started normal distribution and cannot get the ans to this part. Q- The lengths of videos of a certain popular song have a normal distribution with mean 3.9 minutes. 18% of ...
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1answer
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exercise on multivariate normal random variables

The following problem is from Tijms's Understanding Probability. The annual rates of return on the three stocks A, B, and C have a trivariate normal distribution. The rate of return on stock A ...
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Singular Values for Gaussian random matrices

Suppose $X$ is random matrix with rows ${X_1^T, X_2^T, \ldots X_N^T}$ where each of the $X_i$ is an independent random vector with the Gaussian distribution $\mathcal{N}(0,K)$ ($K$ being the ...
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Trouble with seemingly extremely simple statistics question involving normal distribution and expectation.

Say we have a random variable X which is distributed like such: X ~ N(1, 4). A question asks me to calculate $E(X^2)$, which I thought would be straight forward. I use the formula for Variance: $Var(...
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Distribution of linear combination

Let $(X_1,\ldots, X_n)$ be non-independent random variables such that $$\sum_{i=1}^{n} X_i\sim\sum_{i=1}^{n} \alpha (\mathcal{N}(0,1))^2$$ where $\mathcal{N}(0,1)$ is a standard-normal distribution (...
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Probability question : Normal distribution

An auto parts company, produces cylinder liners for engines of 1.2 inches in average diameter with a standard deviation of 0.1 inches. Every piece has a diameter less than an inch or more than 1.4 ...
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Computing the expectation of min and max of two iid gaussian variables

This might seem very easy but i have a question regarding the computation of $\mathbb{E}(X_{(2)} - X_{(1)})$ where $X_1,X_2 \sim \mathcal{N}(\mu,\sigma^2)$ iid and $X_{(1)} = \min\{X_1,X_2\}$, $X_{(2)}...
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What is $E\left[\max(X_1,X_2,X_3)-\min(X_1,X_2,X_3)\right]$ for i.i.d $X_i\sim N(0,1)$?

I have found the distance between the MAX and MIN of 2 random variables in a standard normal distribution. $\text{Distance}=\mathbb{E}|X_1 - X_2|$, where $X$ has a mean of $\mu$ and a variance of $\...
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How do I find the probability of interference of a shaft and a bearing given their nominal diameter values and their standard deviations?

The exact question is as follows: An assembling of shaft and bearing is made out of shaft manufactured to a specification of 30.00 ± 0.09 mm and bearings are manufactured to a specification of 30.10 ±...
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Is square of difference between variables bounded by square of difference between corresponding means?

Assume we have two variables $f_i$ and $f_j$, both of which follows Gaussian distribution, with corresponding means $\mu_i$ and $\mu_j$. Can we get to the conclusion that: ||$\mu_i$ - $\mu_j$||$^2_2$ ...
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1answer
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area under curve must be 1, function intersects with y-axis above 0

what is the best way to find a function that looks like a normal distribution, when the curve intersects the y-axis above 0 (say 0.3) and the area must be 1 (100%)? https://i.stack.imgur.com/5cJSy....
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Let X be$Gamma(\alpha, \lambda)$Prove $(\lambda X - \alpha)/\sqrt{\alpha} \xrightarrow{d} N(0,1)$ as $\alpha \rightarrow \infty$ and $\lambda$is fixed [closed]

First of all the continuity lemma is stated as follows: Let $\mu_n, n=1,2, \dots$ be a sequence of distributions, and $\varphi$ the associated characteristic function. If $\mu_n \xrightarrow{w} \mu$,...
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Spherical Gaussian MLE

I am having trouble doing a derivation. I want to find the MLE estimate of $\sigma^2$ in a spherical gaussian, i.e when we have set $\Sigma = \sigma^2I$. I have already seen https://stats....
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Computing covariance of two standard normal random variables

I'm trying to find the covariance $\text{Cov}(X,Y)$ where $X$ and $Y$ are both standard normal variables. If $\text{Cov}(X,Y)=\Bbb E[XY]-\Bbb E[X]\Bbb E[Y]$, and $\Bbb E[X]\Bbb E[Y]=0$ (both ...
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Deriving the cumulative distribution function for the log-normal distribution

Let $X$ be a random variable having a normal density and consider the random variable $Y = e^X$. Then $Y$ has a log normal density. Find this density of $Y$. If $Z$ is the standard normal ...
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Is the area of a Gaussian proportional to its standard deviation (std)?

Let's suppose that we have a curve that is a Gaussian PDF (probability density function) normalized to an area of one. That is a normal distribution. If we take that gaussian but change it's std to ...
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Joint PDF and Covariance

$X$ is a standard normal random variable and $Y$ is a random variable which takes only the values of either 1 or -1, and $\Bbb E[Y]=1$. $X$ and $Y$ are independent. What is the distribution of $Z=XY$ ...
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normal distribution sample mean

Kofi owns a cinema. He wishes to increase attendances and so considers offering customers unlimited amounts of free popcorn and soft drinks. He estimates that the likely increase in attendances would ...
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1answer
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Self similarity of fractional Brownian motion

The fractional Brownian motion with Hurst Paramter $H\in(0,1)$ is a Gaussian Process $\{X(t),t\ge0\}$ with mean $0$ and covariance function $$\gamma(t,s)=1/2(|t|^{2h}+|s|^{2H}-|t-s|^{2H}).$$ I want ...
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Multiplying a normal distribution by a log-normal distribution

I need direction to approximate the resultant probability distribution of the product of two independent distributions: $N(\mu, \sigma^2)$ and $lognormal(\mu_{N}, \sigma_{N}^2)$, where $\mu_{N}$ is ...
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Bivariate-Normal Conditional Expectation

$X$ and $Y$ are iid standard normal random variables. Assume $a, b, c, d$ and $u$ are constants. Calculate $E( cX + dY | aX + bY = u)$
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Maximum of a Gaussian random vector

Suppose that $X\sim N(0,\Sigma)$ and $Y\sim N(0,\Omega)$ are independent random vectors in $\mathbb{R}^d$ ($\Sigma$ and $\Omega$ are positive-definite). It is known that $$ \mathsf{P}(X\in A)\ge \...
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Leibniz integral rule and change of variable

I want to use the Leibniz rule to calculate the derivative of $J(p)$ in order to p. I do not understand why the derivative will affect only normal: $N\bigg[\frac{u\sqrt{\rho}+N^{-1}(p)}{\sqrt{1-\rho}}\...
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Joint distribution of a set of normal variables.

I am trying to find the joint probability distribution of four variables. These variables are all separately normally distributed with mean 0 and variance 1. I've been looking at various correlation ...
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What is $P\left(|\bar{x}-\mu|\leq 1.5\frac{0.2}{\sqrt{16}}\right)$ equal to?

I'm having a question regarding normalizing the distribution in 2-tailed test. I have $\mu=3.3$, $\sigma = 0.2$ and $n=16$. I need to determine $$ P\left(|\bar{x}-\mu|\leq 1.5\frac{0.2}{\sqrt{16}}\...
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Why do Least Squares Fitting and Propagation of Uncertainty Derivations Rely on Normal Distribution

In learning more about the Normal distribution, I came upon the following sentence in the first section of the Normal Distribution article on Wikipedia Moreover, many results and methods (such as ...
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How to calculate this integration related to normal distribution? [duplicate]

I want to solve a integration of the following expression: $$ \log{P\left(T\right)} = \int_t^{t+T} \log \left[ 1 - \frac{1}{\sigma_{\tau} \sqrt{2\pi}} \exp\left(-\frac{x^2}{...
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How to solve this problem related to normal distribution? [closed]

Given two independent and identically distributed Gaussian random variables $X$ and $Y$. How to calculate the probability that these two random variables are not equal for a period of time $T$? Here, ...
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What is the difference between confidence interval and the width of confidence interval?

I'm having trouble understanding the difference between confidence interval and the width of confidence interval as given in the following question- I solved the first part of the question quite ...
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Find the distribution of $r_{t+2}$ in $AR(1)-ARCH(1,1)$ model

In an $AR(1)-ARCH(1,1)$ model as following: $$r_t=\theta r_{t-1}+u_t$$ $$u_t=\sigma_t\epsilon_t$$ $$\sigma_t^2=\omega+\alpha u_{t-1}^2$$ Where $-1\lt\theta \lt1,\omega\gt0, \alpha\in(0,1)$, and $\...
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Integral of Gaussian Products

In a paper I'm reading, this comes up without further steps: $$ \begin{aligned} \int q_{\boldsymbol{\theta}}(\mathbf{z}) \log p(\mathbf{z}) d \mathbf{z} &=\int \mathcal{N}\left(\mathbf{z} ; \...
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How to calculate this integration about normal distribution?

I want to solve a integration of the following expression: $$ \log{T} = \int_t^{t+T} \log \left[ 1 - \frac{1}{\sigma_{\tau} \sqrt{2\pi}} \exp\left(-\frac{x^2}{2\sigma_{\tau}^...
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Normal Distribution same p values for different Z

Question I was doing this question (I know how to do it) and at some point I needed to find the inverse normal of 0.9994 (standard normal distribution). I'm allowed statistical tables (z to 2 d.p) ...
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Calculate derivative of partial effect in probit with respect to the parameter

In the standard probit regression we have that $ Pr(y>0| X) = \Phi(\frac{X\beta}{\sigma})$. With $\Phi$ Cumulative Density Function of the Normal Distribution. The marginal effect of the ...
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Normal distribution non linear transformation

I have the following problem : Given $X \sim N(\mu,\sigma^2)$ and $X' = h(X) = (\frac{x-\mu}{\sigma})^2$ Find $E[X']$ and $V[X']$. My reasoning is as follow : Since $X' \sim (\frac{x-\mu}{\...