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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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Indefinite integral involving the normal distribution and its CDF times x^2

I'm trying to evaluate the integral \begin{equation} \int_{-\infty}^{\infty} s^2 f_{s_1} (s) \Phi \Big(\dfrac{s}{\sigma_n} \Big) \, ds. \end{equation} where \begin{equation} f_{\bar{s} _l}(s) = \...
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1answer
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Estimating temperature most accurately with different thermometers

I recently came upon this semi-opened ended question and wanted to think through it with you guys. You have 5 measurements from 5 different thermometers, which are unbiased, but each with a ...
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1answer
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Can we compute the infinite series of the cdf of a standard normal distribution $\sum^{\infty}_{n=0} \Phi(-\sqrt{n})$?

I've tried plugging the series in Wolfram to first of all check if it converges but it doesn't return anything useful. Moreover, just computing $\Phi(-100)$ returns something useless. Looking at the ...
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1answer
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When can i use a normal distribution to describe my data?

I have dataset which is markedly left-skewed, and I wonder if it will be inappropriate to use the normal distribution curve to analyses the data given it's highly skewed? Picture of my data
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1answer
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Linear transformation of multivariate normal distribution to a higher dimension?

Suppose I have transformation defined as $Y_{q\times 1} = C_{q\times p}X_{p\times 1}$, where $X \sim N_p(\mu, \Sigma)$. If $q > p$ how do I compute the distribution of $Y$, since I think the ...
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1answer
23 views

Convergence to normal distribution without i.i.d.

Suppose that $X_n$ are independent and $\mathbb{P}(X_m = m) = \mathbb{P}(X_m=-m) = \frac{1}{2m^2}$ and for $m\geq 2$ \begin{equation*} \mathbb{P}(X_m = 1) = \mathbb{P}(X_m=-1) = \frac{1-m^{-2}}{2}. \...
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1answer
19 views

Joint density of two vectors of multivariate normal random variables

If $\bf{X}$ and $\bf{Y}$ are dependent multivariate normal random variables, what is the joint density of $\bf{X}$ and $\bf{Y}$? Is it also multivariate normal?
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STATISTICS HELP [on hold]

we found that weights of adult green sea urchins are normally distributed with mean of 52 g and standard deviation of 17.2 what is the percent of adult green sea urchins that weigh above 40g? Would I ...
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1answer
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Given a sequence $\{ X_{n} : n \in \mathbb{N} \}$ of standard normally distributed random variables, what is the probability that $X_{1}+X_{2}<1$?

I'm looking for a general formula that can be used to compute the probability that a sum of standard normal random variables is above a certain constant $a$. So for example the probability that $X_{1}+...
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Convergence related to normal cdf

Consider the following problem: For each $x>0$, let $y\in\mathbb R$ solve the equation $$\int_y^{y+x}[\Phi(z)-r]dz=0,$$ where $r\in (0,1)$, and $\Phi(\cdot)$ is the normal cdf. We can show that ...
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1answer
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Conditional marginal distribution of conditional bivariate normal distribution

I have a bivariate normal distribution$$(X, Y)\sim N(\mu_{x}, \mu_{y}, \sigma_{x}^2, \sigma_{y}^2, \rho)$$ My question is : when $X > k$ ($k$ is a constant),how to get the distribution of $Y$? Can ...
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Polar form of normal random vector , angle and length are independent ,and angle is spherical distribution

Represent $g \sim N(0,I_n)$in polar form as $g=r \theta$ where $r = \|g\|_2$ is the length and $\theta = \frac{g}{\|g\|_2} $ is the direction prove that $r$ and $\theta$ are independent ? prove ...
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1answer
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How to calculate sample mean and variance given the confidence interval for the normal?

A random sample of size n=16 is taken from a random variable X~N(mu, sigma), with variance unknown. The 95% confidence interval for mu (44.7, 49.9). What are the values of the sample mean and the ...
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Probability econometrics

Solve let B distributed as N (0,I) and consider the linear transformations Y= b + BX, where b is a vector and B a k×n matrix of constants, and Z=c+CX where c is an m×1 vector and C an m×n matrix of ...
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1answer
23 views

Expected value of the product of dependent Normal random variables

I have 3 independent Normal Random Variables: $A$, $B$ and $C$, each with mean=$0$ and Variance $1$. Then I have $X=3A+5B$ and $Y=A-C$... because both of them are functions of $A$, we know they ...
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1answer
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Simplify double sum

Does the following expression has a closed form \begin{align} E \left[ \| Z\|^k \exp(t \|Z\|^2)\right] \end{align} for $k$ is even and $Z$ is standard normal vector. For the case of $k=2$ the ...
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1answer
21 views

Determining sample size given true proportion.

I'm attempting to solve a problem from a statistics course in regards to finding the sample size I need to take when given the Margin of Error, Confidence interval, and 'true proportion' (probability)....
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Random variable $X \sim \mathcal N(m,m^2)$ question

A random variable $X \sim \mathcal N(m,m^2)$ in a population, with $m \in \Bbb R^+$. In what percentage of the population, the variable has a positive value?. I don't know how to start this problem, ...
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2answers
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Statistical Integral in Financial Mathematics [duplicate]

I need to show that $$\frac{1}{\sqrt {2\pi}} \int_{-\infty}^{+\infty} x^{2n} e^{\frac{-x^2}{2}}dx = (2n-1)!!$$ Integration by parts seems to be the best apporach but I cannot seem to figure my way ...
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1answer
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$\mathbb{E}(X_{Y+1}X_{2}^{2}X_{2}|x_{1})$ with $X\sim N(0,1)$ and $Y\sim Pois(1)$ both independent

Let $\{X_{i},i\in\mathbb{N}\}$ be a sequence of independent standard normal random variables. Furthermore, $Y$ is a Poisson distributed random variable with parameter $\lambda=1$, i.e., $\mathbb{P}(Y=...
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1answer
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Questions on the Mathematics behind Expected Shortfall and Value at Risk

I have a couple questions on the mathematics behind some general value at risk (VaR) and expected shortfall (ES) calculations. $$ P(L > VaR(\alpha))=\alpha $$ $$ ES(\alpha)=E[L|L > VaR(\alpha)] ...
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1answer
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KL divergence between Exponential and Normal distributions

What is the KL divergence $KL(P \;\Vert\; Q)$ between an Exponential distribution $P = \text{Exp}(\lambda)$ and a Normal distribution $Q = \mathcal N(\mu, \sigma^2)$? I have not found any source for ...
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2answers
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Showing $\sum a_iX_i$ and $\sum b_iX_i$ are independent iff $\sum a_i*b_i=0$ where $X_i's$ are i.i.d N(θ,σ2)

Let $X_1, X_2,\ldots, X_n$ be i.i.d with the distribution $N(\mu, \sigma^2)$. Prove $Y =\sum_{i=1}^{n} a_iX_i$ and $Z =\sum_{i=1}^{n} b_iX_i$ are independent iff $\sum a_ib_i=0$. I have proved it by ...
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1answer
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Sum of square product of I.I.D. Normal distribution

All $a_i$,$b_i$ and $c_i$ $(1<i<10)$ are independent standard normal distributed variable by each other ($a_i$,$b_i$,$c_i$$∼$$N(0,1)$). and I would like to know about expected value of ...
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1answer
63 views

Integral of pdf and cdf normal standard distribution

$$ \int_{-\infty}^{\infty}\Phi(a+bx)\phi(x)dx=\Phi\left(\frac{a}{\sqrt{1+b^2}}\right) $$ I have a problem with showing the above result, where $\phi(x)$ and $\Phi(x)$ respectively are the pdf and cdf ...
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Error probability of a digital code through noisy channel

Suppose a binary message is transmitted through a noisy channel. The transmitted signal $S$ has uniform probability to be either $1$ or $−1$, the noise $N$ follows normal distribution $N(0,4)$ and the ...
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Conditional expcetation of a function of multivarite normal random variables

Say we have some function of bivariate standard normal random variables $f(x_1,x_2)$ which we approximate with first 2 terms of its Taylor series, $$z = f(x_1,x_2) \approx f(0,0) + f_1(0,0)x_1 + f_2(...
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1answer
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Calculating probabilities in normal distribution

If $X$ is a normal random variable with $\mu=-2$ and $\sigma=3$ , and has probability density function $f_x$ and cumulative density function $F_x$ , calculate- $1)$ $P(-3<X<0)$ $2)$ $F^{-1}(1/...
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Relating posterior to the least square estimator of W

I'm currently working on an assigment and I'm currently stuck and could really use some help, I've been given the fact that my prior over my parameters W is given by a gaussian pdf, likewise is the ...
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1answer
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Prove the sampling distribution of $S^2$ has the mean $\sigma^2$ and the variance $2\sigma^4/(n-1)$

I would like to ask whether anyone would mind providing me with some direction on how to proceed with this proof. The question asked me to use the theorem below to prove that, for random sample of ...
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1answer
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Is a random vector multivariate normal if and only if every linear combination of its coordinates is normal?

The first sentence of this Wikipedia article (https://en.wikipedia.org/wiki/Gaussian_process) seems to imply that a vector of random variables is multivariate normal if (and only if) every linear ...
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Mixture distributions [closed]

X and Y are independent and normally distributed with means $u_x,u_y$ and std.deviations $\sigma_x,\sigma_y$ I define two new random variables: $$A= \frac{1}{2} (X+Y) $$ and $B$ such that its density ...
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2answers
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Reconstructing normal distribution according to data ranges

I have a temperature data and I believe it follows the normal distribution. The problem is that I know just values for few ranges, but I need to have the results for finer temperature classes. So, as ...
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1answer
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Using the normal distribution to approximate the binomial distribution (weird example)

The problem: suppose 406 dice are rolled. Find the probability of getting exactly 100 5s. Binomial definitely applies, since there is a fixed number of trials (406 dice) and there are two distinct ...
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1answer
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Expected value of $[(X-\mu)/\sigma]^2$

I would like to ask that, there is a question asking to show that $\bar{X}$ is a minimum variance unbiased estimator of the mean $\mu$ of a normal distribution. I am having difficulty understanding ...
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Put an expression under a normal form.

I am struggling with the following problem : I have the expression of $$\mathrm{log(p(\theta|y, \alpha, \beta)) = C - \frac \beta2 \|y-\Phi\theta\|^2} - \frac \alpha2 \|\theta\|^2 $$ with $C$ a ...
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2answers
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What is the reason that Student-t Distribution is used when the number of samples is small

Let $\bar{X}$ be the distribution of sample mean for $n$ identical and independent distributed as Normal distributions $N(\mu, \sigma^2)$. The random variable $$ \frac{\bar{X} - \mu}{\frac{\sigma}{\...
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1answer
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Two different answers when integrating with respect to Brownian motion

Consider the integral with respect to Brownian motion $$\int_{0}^{t}s \ dB_{s} \ . $$ A textbook I am reading uses integration by parts to rewrite the above integral as $$tB_{t}-\int_{0}^{t}B_{s} \ ...
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1answer
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Statistical distribution of a product of lognormal variables

Question I have two correlated lognormal variables $S_1$ and $S_2$, such that $$\begin{pmatrix} \ln S_1 \\ \ln S_2 \end{pmatrix} \sim \mathcal{N}\left( \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix}, \...
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Generating vector inside a $n$-sphere

I want to generate k n-dimensional vectors which are all inside a r-radius n-sphere and the most important : I want something uniformly distributed inside the n-sphere. My initial idea is to generate ...
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Derivatives of Gaussian measure with respect to (co)variance

For $a > 0$, let $G_{a}$ denote a centered Gaussian random variable with variance $a$. That is, the density of $G_a$ is $\frac{e^{\frac{-x^2}{2a}}}{\sqrt{2\pi a}}.$ Fix $t > 0$ and define a ...
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Test hypothesis, probability of rejecting $H_0$

A mechanic wants to test how difficult it is to unscrew the bolts that hold the tires in place in a car. Assume that out of $8$ cars tested, the average force was $259.634$ newtons. Also assume that ...
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Independence of a multivariate gaussian

It can be shown that a multivariate gaussian with diagonal covariance matrix can be factorized into a product of univariate gaussians, which means that the variables are independent (with independent ...
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1answer
28 views

expectation & variance of square of non-standard normal distribution

Let $A \sim Normal(k, 1)$ (ie. with mean $k$ and variance $1$). Let $B = A^2$. The first step is to express the CDF of $B$ in terms of the CDF $S_X(x)$ of the standard normal distribution (with mean $...
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(Generic)Find out number of ways to distribute 'c' number of chocolate in 'p' number of person where no person can have more than 'n' chocolate

(Generic)Find out number of ways to distribute 'c' number of chocolate in 'p' number of person where no person can have more than 'n' chocolate. Note:person can have 0 (zero) chocolate. eg c = ...
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Probability that more than 220 heads appear in 400 coin flips

If I toss 400 fair coins, estimate (to the nearest whole percentage point) the probability that more than 220 heads appear. Here is my thought process: This is a binomial variable. We know that $p = ...
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Product of Pareto RV and Normal RV : PDF and other characteristics

I am struggling to find the PDF of a product of two random variables. $ a = x \times h$ $ x \sim \mathcal{N}( \mu, \sigma^2 )$ $ h \sim \mathcal{P} (\alpha, x_m)$ I know the PDF's of the two ...
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Verifying a Gaussian Process

Suppose I have two independent standard normally distributed random variables $X_1$ and $X_2$ and form the linear combination of these to define a random process {$Y_t$} by; $$Y_t=X_1\,\text{cos}(t)+...
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Why does standardizing normal distributions preserve probabilities?

Normalizing a normal distribution to the standrad normal distribution is achieved by creating from a random variable $X$ a new random variable $X' = \frac{X - \mu}{\sigma}$, where $\mu$ is the mean of ...