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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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Poisson distribution to normal distribution?statistics

I need to demonstrate why when (lambda)is big enought poisson distribution becomes (aproximation)to normal distribution. Thanks you
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1answer
23 views

Why a normal distribution would not give a good approximation to a specific discrete distribution?

I have a discrete distribution $X$ with mean $56.87500$ and standard deviation $70.725$. I also have that the support of $X$ is contained in $[0, 8750]$, the maximum value of $X$ is $8750$ and $p(X=0)=...
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9 views

How to vectorize/matricize multivariate Gaussian PDF for more efficient computation?

Context: I was recently implementing (in Python) the Expectation-Maximization (EM) algorithm for Gaussian mixture models, and part of that process involves computing the Gaussian PDF for various ...
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2answers
25 views

Find the probability of a Normal Distribution random variable

The excercise is given as it follows: Let the Temperature $T$ during a month of a year has a normal distribution with mean $68°$ and a standard deviation of $6°$. Find the probability $p$ that the ...
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1answer
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What are mean and variance of $W_i$, given that $Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1)$? [on hold]

Let $$Z_n=\frac{\sum{W_i}}{\sqrt{n}\sigma}\sim N(0,1),$$ where $W_i=X_i-\mu$. What are the mean and variance of $W_i$?
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17 views

Distribution function of X-Y for normally distributed random variables [duplicate]

I have two independent normally distributed random variables $X,Y\sim \cal N(\mu,\sigma^2)$ and want to calculate the distribution of $X-Y$. I tried with $F(z)=P(X-Y \leq z)$ but failed. Does anyone ...
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15 views

Why find the inflection points on a bell curve?

I understand that the inflection points on a standard normal curve can be calculated with the equation $\mu \pm \sigma$, or essentially where z = $\pm$1. I don't fully understand the significance of ...
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1answer
23 views

Statistics and Confidence Intervals

Given the following set of values: 10,11,14,95,73,30,29,9,97,94,70 How do I calculate a 99% confidence interval for the sample mean? I am assuming that the variance is 10 Well, the idea I have is ...
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24 views

How far can the mean of a normal distribution be from its arithmetic mean?

Let the arithmetic mean of the first $n$ terms of the sequence $S_n$ be $\mu_n$. Further assume that $0 < S_n < a$ for all $n \ge 1$. If $S_n$ is normally distributed with a mean $\mu_d$ and ...
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0answers
9 views

relating integrals of normal distribution

I recently came across the following problem. Unfortunately, my statistics knowledge is a little rusty, so I hope that someone here can help me: Let $f(x \mid \mu, \sigma^2)$ be a normal ...
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1answer
27 views

Mean preserving spread for normal distribution

Is it true that any two normal distributions with the same mean can be ordered w.r.t. the relation of a mean preserving spread? My intuition would be that this is true but I cannot come up with a ...
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10 views

Integral of logistic normal distribution approximation

Following paper about Glicko rating have a expression below: Parameter estimation in large dynamic paired comparison experiments (Equation 18, 19 on Appendix A) $$ \int \frac{ (10^{(\theta-\theta_j)...
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22 views

Computing the PDF of a low-rank multivariate normal

I have a question which seems simple, but I would appreciate some comments! Sometimes models involve a low-rank approximation to a covariance matrix. What confuses me is how you can compute the PDF ...
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1answer
31 views

Sufficient statistic for class of distributions

For the class $\{F_{\theta_1}, F_{\theta_2}\}$ of two DFs where $F_{\theta_1}$ is $N(0,1)$ and $F_{\theta_2}$ is $C(0,1)$, find a sufficient statistic. Let, $X_1, X_2, \dots, X_n$ is a random sample ...
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14 views

Central Limit Theorem with linear transformation?

Suppose $Z=(z_{1},...,z_{m})^T$ is m dimensional vector, each $z_{i}$ is independent identical distribution with mean 0. If we do linear transformation like: $$X = \Gamma Z$$ where $\Gamma$ is $p*m(p&...
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1answer
40 views

Condition Expectation of normal variables

Let X,Y be jointly normal. Then I know that $E(X|Y)=E(X)+\frac{Cov(X,Y)}{V(Y)}(Y-E(Y))$. Do I need joint normality for this result? Does it also hold, if just X is normal and Y is normal?
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Degenerate Bivariate Normal [duplicate]

Let $X$ be normal. Is $(X,b)$ (a random vector in $\mathbb{R}^2$), where $b$ is just a constant, degenerate bivariate normal? Can I say that? Thanks!
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1answer
27 views

Linear Regression Assumption: Normality of residual vs normality of variables

I have read in many places, including stack exchange, that in order to carry linear regression analysis the residuals have to be normal. This is required because most of the statistical results, ...
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19 views

Simplify equation - How to normalize log probability and turn it back into its raw probability?

I am having trouble simplifying this equation below. It computes a probability of a Gaussian over two products. To do this efficiently, I am taking the log of the probability to change the products ...
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0answers
20 views

Calculating inverses in probability

I am asked to find the number of proposals to submit such that at least one proposal is approved with at least $99\%$ probability. I have found that the probability of at least one proposal being ...
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0answers
21 views

Mean and Median are equal. Find x and Y. X and Y are natural numbers. [on hold]

4,6,x,y,10. The mean and median are equal to each other. Find x and y?
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11 views

How to compute $cov(\frac{1}{2}Z'AZ,\frac{1}{2}Z'BZ)$?

I want to show that $cov(\frac{1}{2}\textbf{Z}'A\textbf{Z},\frac{1}{2}\textbf{Z}'B\textbf{Z})=\frac{1}{2}tr(AR(\theta)BR(\theta))$ where $A,B \in Sym(p)$ (real symmetric $p \times p$ matrices) and $\...
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2answers
19 views

Hypothesis testing for sample means within a normal distrubution

The context of the question is that a bakery bakes cakes and the mass of cake is demoted by $X$ such that $X \sim N(300, 40^2)$. A sample of 12 cakes is taken and the mean of the sample is 292g. The ...
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promotional sample data follow normal distribution

I have a question about proportional sample data's distribution. When assuming p is the probability of success in sample data, and 1-p the failure, some sites suggested when np>10 and n(1-p)>10 then ...
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3answers
102 views

Is $(a+bX,X)$ jointly normal when $X$ is normal?

Let $X$ be a normal random variable and $Y=a+bX$, where $a,b$ are just some constants. Then, is it true that $(Y,X)$ are jointly normal? If yes, how can I easily see that? Thanks!
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1answer
38 views

Distribution of $X(t)=A\cos(t)+B\sin(t), t \ge 0$. $A,B$ iid $\sim N(0,1)$

Let $A, B$ be independent random variables which are both $N(0,1)$ distributed. $X(t)=A\cos(t)+B\sin(t), t \ge 0$ is a stochastic process. I want to determine a) $\mathbb E[X(t)], \mathbb V[X(...
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1answer
26 views

How can $\sigma^2$ be derived as a function of $\mu$ in a Gaussian pdf?

I have a Gaussian pdf defined as $$f_X(x) =\frac{1}{\sigma\sqrt{2\pi}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}$$ whose $\mu = \frac{d^2}{6D}$, where $d$ is distance parameter and $D$ is the ...
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0answers
8 views

Hamiltonian monte carlo sampling : Energy Histogram vs Sample Histogram

I'm using HMC to sample from an N-d Gaussian. So the PDF is that of a multivariate normal distribution. HMC requires an energy function and its gradient. The library I'm using, maximizes the ...
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1answer
11 views

From randn to bivariate Gaussian distribution image

In Matlab, I generated a bivariate Gaussian distribution with mean 50 and standard deviation 50: G=50*randn((100*100)/2,2)+50; histogram2(G(:,1),G(:,2)); The 3D ...
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P(X+Y<a | X<a) for X, Y normal and independent distributions

let $a\in \mathbb{R}$ and let X and Y be independent normally distributed random variables, with mean $0$ and respective variances $\sigma^2_X$ and $\sigma^2_Y$. Can we express $$P(X+Y<a\,|\,X<a)...
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0answers
20 views

Solve loss function for a normal distribution by integration

I want to compute the loss function for a normal distribution with mean $\mu$ and standard deviation $\sigma$. I need this for an inventory optimization model as basically I want to know "If I ...
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0answers
5 views

Retrieving a matrix from its Schur complement

I came across a problem that pertains to Schur complements and Gaussian conditional law. Consider $x \sim \mathcal{N}(0, \Sigma_{xx})$ an $n$-variate Gaussian random variable and an independent (of $x$...
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1answer
16 views

integration of a Gaussian pdf

Where $m_x$ and $\sigma_x$ are respectively the mean and standard deviation of $x(t)$, given an exact expression for $f_y(y)$: $$ f_y(y)=\frac{1}{3(2\pi)^{1/2}\sigma_xy^{2/3}}\text{exp}(-\frac{(y^...
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Does the Johnson–Lindenstrauss lemma require the Normal distribution?

The Johnson–Lindenstrauss lemma states that a set of points in a high-dimensional space can be embedded into a space of much lower dimension in such a way that distances between the points are nearly ...
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1answer
34 views

Proving that $\Gamma(\frac{1}{2})=\sqrt(\pi)$ using the expected value of standard normal variable (integral calculation)

I'm looking to prove that $\gamma$$(\frac{1}{2})=\sqrt(\pi)$ using the fact that $E(Z^2)=\int_{-\infty}^{\infty} \frac{1}{\sqrt(2\pi)}e^{\frac{-z^2}{2}} z^2 dz$ (where $Z$ is a standard normal ...
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1answer
18 views

Conditional Multivariate Normal (beyond the bivariate case) [duplicate]

so I need to obtain the conditional distribution of a multivariate normal. However, I can only find it for the bivariate case: $$(x_1|x_2=a) \sim N(\bar{\mu}, \bar{\Sigma})$$ $$\bar{\mu}= \mu_1 + \...
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0answers
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Real life Cramer's Decomposition Theorem example.

I just heard of CDT and "tried" to read online documents (Not to mention "understand" it). Easy Version Definition: Given independent normally distributed random variables ξ1, ξ2, their sum is ...
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1answer
22 views

Relationship between cdf of normal distribution and uniform distribution defined on $[0,1]$

So the problem is, given $X\sim N(1,2^2)$, $Y=e^X$, if the cdf of standard normal distribution is $\Phi$, to show that $$\Phi\left(\frac{\ln Y-1}{2}\right)\sim U[0,1],$$ where $U[0,1]$ is the ...
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1answer
14 views

Joint distribution for sums of normal variables

Given $n$ independent normally distributed variables $\{\xi_k\}_{k=1}^n$. Each $\xi_k$ has expectation $a$ and dispersion $\sigma^2$. I need to find joint distribution for two variables: $\eta=\sum_{...
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14 views

The value of houses in a certain district is normally distributed with mean $\$235420$ and standard deviation $\$28724$ (see below)

The value of houses in a certain district is normally distributed with mean $\$235,420$ and standard deviation $\$28,724$. a) Find the percentage of houses in this district with values from $\$200,...
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1answer
39 views

CDF of positively correlated Gaussians

Suppose $X,Y$ are two positively correlated Gaussians with zero mean and unit variance. Is it the case that for $a,b \in \mathbb{R}$, $$ \Pr[X \leq a, Y \leq b] \geq \Pr[X \leq a] \Pr[Y \leq b]? $$
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Product of Normal and independent log-Normal. What is the density?

Let $X$ and $Y$ be independent standard Gaussian random variabless -- i.e., $N(0,1)$. Let $\sigma \in \mathbb{R}$. Define $$Z=X \cdot e^{\sigma \cdot Y}.$$ Is there a closed form expression for the ...
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1answer
39 views

Expectation of Maximum of Translated Gaussian

Given $X\propto N(\mu,\psi^2)$ and $K$ constant, find $E(\max\{K-X,0\})$. My attempt: $$ \begin{align} E(\max\{K-X,0\})\\ &=\int_{-\infty}^{\infty}\max\{K-x,0\}f_X(x)dx\\ &=\int_{K}^{\infty}(...
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1answer
32 views

Estimating the standard deviation based on sample

I have been given a problem for homework with which I need a little help. In part A, I suspect that the solution has to do something with the degrees of freedom where we use n-1 for sample and n for ...
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1answer
24 views

Normal interarrival times

I have the interarrival time $t$ follows a normal distribution of $N(8,4)$. I'm trying to find $P(t < 0)$ The probability that the 16th and the 9th customer arrive within 55 minutes of each other ...
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1answer
20 views

Probability of selling stock

I have 4 stocks $x_1$, $x_2$, $x_3$, $x_4$ where $x_i$ ~ $N(\$5000,2500)$ What should be the probability of me selling all 4 stocks when At least one stock exceeds \$5022 All four exceed \$5022 The ...
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1answer
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The ideal size of a class of a college is 150.

The ideal size of a class of a college is $150$ students. The college, experienced from past, knows that only $30$% of the admitted students will actually attend. The college uses a policy of ...
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Expectation of product of normal CDFs

Suppose $X \sim N(0, I)$. How would I go about calculating \begin{align*} \mathbb{E}[\Phi(\Delta_1 + \theta_1^\intercal X)\Phi(\Delta_2 + \theta_2^\intercal X)]? \end{align*} where $\Delta_1, \...
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25 views

Determine the distribution of $\Theta$ (Finding the range)

Suppose that $X$ and $Y$ have a two-dimensional normal distribution with means $0$, variances 1, and correlation coefficient $\rho$, $|\rho| < 1$. Let $(R, \Theta)$ be the polar coordinates. ...