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

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Bayesian Estimation of the mean of a multi-variate Gaussian

The posterior mean of a multivariate normal distribution is to be estimated with the Bayes rule for densities (http://www.math.uah.edu/stat/dist/Conditional.html), following the approach as described ...
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14 views

Normal Distribution finding q Question

The random variable X is normally distributed with mean 82 and standard deviation 7.4. Find the value of q such that P (82−q < X < 82+q) = 0.44. My friend sent me this answer. My doubt ...
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1answer
10 views

Variance of not quite the product of two independent, normally distributed random variables

Let's say I have two independent variables, $X\sim N(10,9)$ and $Y\sim N(5,4)$. $X$ represents the number of orders received in a month, and $Y$ represents the size of each order. For this example, a ...
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0answers
11 views

Expected value of norm of multivariate normal distribution random vector

Let $X$ is a random vector size $p$ from multivariate normal distribution $\mathcal{N}$($0$, $\sigma$ $I$), $I$ is identity matrix. I want to find the expected value of reciprocal of norm like this ...
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42 views

Convert min to max probability

Assuming $Y=min(w_1,\ldots,w_n)$ , $w_i\sim N(\mu,\sigma^2) i.i.d$ I want to express $Y$ in terms of the $Q$ function. Knowing that $Y=min(w_1,\ldots,w_n)=-max(-w_1,\ldots,-w_n)$ $P(Y\leq y)= ...
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1answer
34 views

What distribution it is based on the histogram? [on hold]

I generated this histogram in r and was trying to determine which distribution I should use, my guess is normal or Binormial. But I'm not sure, can anyone help please?
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1answer
27 views

Product of two multivariate Gaussian pdfs - normalization constant

https://www.cs.nyu.edu/~roweis/notes/gaussid.pdf contains expressions (p.2, 6e, 6f) for the normalization constant for the product of two multivariate Gaussian pdfs, with mean vectors $a$ and $b$ ...
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1answer
33 views

Kolmogorov-Smirnov two-sample test

I want to test if two samples are drawn from the same distribution. I generated two random arrays and used a python function to derive the KS statistic $D$ and the two-tailed p-value $P$: ...
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10 views

What is the distribution of 'max of some normaldistributions'?

Suppose I have two random variables $a$ and $b$. $a$ follows a normal distribution of parameters $u_1, s_1$. $b$ follows a normal distribution of parameters $u_2, s_2$. $u_1$ and $u_2$ are the ...
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1answer
23 views

How to approximate a normal distribution?

Suppose I have two random variables $a$ and $b$. $a$ follows a normal distribution of parameters $u_1, s_1$. $b$ follows a normal distribution of parameters $u_2, s_2$. $u_1$ and $u_2$ are the ...
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7 views

Can I use Geometric Distribution to find the law of a total?

I have a variable X which is the amount of minerals in a dL(deciliter) of water. X follows a Normal Distribution X~N(μ,σ). I have the probabilty of the P(a ≤X< b) in a dl, where a and b are ...
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19 views

Is there any closed form for the integration of multiplication of two multivariate normal probability distributions?

I already computed the following integration but its a messy thing. I wonder if there is any easy way to compute it? or it has any closed form? V and p are known where V and p (p<1) are positive. ...
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1answer
10 views

Variance with minimal MSE in normal distribution

Given $X_1,...,X_n$ ~ i.i.d. $N(\mu, \sigma^2)$ where the mean is unknown, let the estimator for $\sigma^2$ be $\hat{e} = p\sum_{i=1}^n(X_i-\overline{X})^2$ How do I choose $p$ so that this estimator ...
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0answers
17 views

First order moment of multivariate Gaussian random vector

Let $X = (X_1,\dotsc, X_n)$ be a random vector distributed as a multivariate Gaussian with mean $0$ and covariance $\Sigma$. What is $\mathbb{E}[X_1\dots X_n]$?
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2answers
19 views

Covariance between $X$ and $Y$ of a bivariate normal distribution?

$X$ and $Y$ have a bivariate normal distribution with $\sigma_X$= 5 mL, $\sigma_Y$= 2 mL, $\mu_X$= 120 mL, $\mu_Y$= 100 mL, and $\rho$ = 0.6. How do I find the covariance of $X$ and $Y$? I know the ...
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24 views

Property of covariance of Normal random variable with an arbitrary function of that random variable

In the paper Sharpee, T., Rust, N.C., Bialek, W.: Analyzing neural responses to natural signals: maximally informative dimensions. Neural Comput. 16, 223–250 (2004). I found the following claim ...
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29 views

If B is a N(0,1) R.V., show $E[B^4] = 3$

I've read in Elementary Stochastic Processes by Mikosch (p. 98), that it is a well known fact that: If B is a N(0,1) R.V., $E[B^4] = 3$ I also see something equivalent (but uncited) on the ...
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0answers
33 views

Expected Value of the absolute value of the sum of random variables

Hi everyone and thanks in advance. Let's say we have a random variable Y which can be expressed as the sum of two other complex random variables X and W, i.e. $ Y = X + W $. $X$ and $W$ are ...
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15 views

Scaled distribution of Brownian motion

If I have $X = 5(B_t - B_s)$ Does this have a distribution of $\sim \text{N}(0,25(t-s))$ ? Since $B_t - B_s$ has distribution $\sim \text{N}(0,t-s)$ Then $X = \mu \cdot 0 + \sigma_1 Z$ where $Z ...
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1answer
20 views

Frequency Distribution and Throughput

I am conducting an experiment on a couple of computer systems but the results I have don't make sense to me. I made each system perform 1000 operations: System A performs operations at a rate of ...
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0answers
28 views

To determine asymptotic value of funtion for large N

To show that the function follows normal Gaussian for large value of N (s.t. m is much less than N ) with mean at 'm'. $f(m,N)=\sum_{a=1}^{\lfloor N/2 \rfloor} \binom{N}{S} * ...
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0answers
29 views

To determine probability distribution for large $N$ with mean at m [closed]

To show that the following expression turns to Gaussian for large value of $N$ $$\binom{N}{S}\binom{X-1}{a}\binom{Y-1}{a-1}$$ where X+Y+S=N. To show it shows normal distribution with mean at 'm' and ...
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1answer
20 views

How to extract a covariance matrix with this information

Referring to the above image, I wanted to know how to get the covariance matrix $\sum$. My understanding is, $A$, is our transformation matrice, such that $\begin{bmatrix} X_1 \\ X_2 \\ ...
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24 views

Product of (multivariate) Gaussian densities

One can frequently read that the product of two multivariate Gaussian pdf f1(x) and f2(x) is itself a Gaussian, with parameters as defined for example in: ...
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19 views

Adding x+y gamma function [closed]

I know how to set up a independent variable when it's normal. And I'm sure they are somehow related but I'm not sure where to start on this problem. It's not covered at all in the book I have and he ...
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1answer
21 views

Linear transforms of Normal dist [closed]

If $X_t = \sqrt{t} Z$ where $Z \sim \text{N}(0,1)$ Then show the distribution of $X_t - X_s$ for $s<t$ Just wanted to check, would this be $\sim \text{N}(0,t-s)$ or $\sim \text{N}(0,(t-s)^2)$ ?
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2answers
24 views

Distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion

I am trying to find the distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion. One (very hand-wavey) way is to assume a priori that it is Normally distributed. Then one can ...
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0answers
8 views

Normality assumption is not met [closed]

. Suppose one hundred samples of size n = 3 are taken from each of the pdfs (1) fY (y) = 2y, 0 ≤ y ≤ 1 and (2) fY (y) = 4y3 , 0 ≤ y ≤ 1 and for each set of three observations, the ratio y − μ s/ √3 is ...
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1answer
22 views

Decision-making with random term

Consider the following situation. There are multiple options to choose from based on an attribute related to those options. For example: ...
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1answer
14 views

Normal Distribution: Statistics

I'm having a lot of trouble trying to remember the formulas on how to calculate these questions. Any help would be great. An automobile insurer has found that repair claims are Normally distributed ...
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1answer
22 views

distribution of distance between two points whose coordinates are normal random variables

let there be two random variables $(X_1,Y_1)$ and $(X_2,Y_2)$, where $X_1\sim N(m_1,s)$, $X_2\sim N(m2,s)$, $Y_1\sim N(n,t)$, $Y_2\sim N(n,t)$. What is the distribution of $\|(X_1,Y_1)-(X_2,Y_2)\|$?
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1answer
37 views

computing p-value with small n

As part of the quality-control program for a catalyst manufacturing line, the raw materials (alumina and a binder) are tested for purity. The process requires that the purity of the alumina be greater ...
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2answers
31 views

Confusion with Z-Score

Having some issue with the concept of Z score. When exactly do I use $Z = \frac{\bar X - u}{\sigma}$, and when do I use Z = $Z = \frac{\bar X - u}{\frac{\sigma}{\sqrt{n}}}$. I get very confused ...
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1answer
54 views

Finding the probability using a normal distrubtion.

I have a stats question that says, "An airline flies airplanes that hold 100 passengers. Typically, some 10% of the passengers with reservations do not show up for the flight. The ...
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1answer
70 views

inequality with gaussian cdf and density involved

in my calculations I've arrived at the following inequality $$ |\frac{4\phi(x)(1-2\Phi(x))}{(1+(1-2\Phi(x))^2)^2}| \leq 0.5 $$ where $\phi$ is Gaussian density, and $\Phi$ Gaussian cdf, which can ...
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1answer
45 views

Calculate P-Value

In a certain area, regulations require that the chlorine level in wastewater discharges be less than 100 $\mu$/L. In a sample of 85 wastewater specimens, the mean chlorine concentration was 98 ...
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2answers
44 views

Finding distribution of distance from origin

A shot is fired at a circular target. The vertical and horizontal coordinates of the point of impact (taking the centre of the target as origin) are independent random variables, each distributed ...
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1answer
55 views

Expectation of a linear combinations of iid standard normal, restricted to a halfspace

Let $u = (u_1, \ldots, u_n)\in\mathbb{R}^n$ be a unit vector in $\mathbb{R}^n$, $Y_i$ be i.i.d standard normal Is there any easy way to calculate $$\mathbb{E} \left[ 1_{\displaystyle \left\{ ...
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28 views

How to fit normal cumulative distribution functions

For a normal distribution $N(\mu,\sigma^2)$, we know its cumulative distribution function is $F(x)=\Phi(\frac{x-\mu}{\sigma})$ where $\Phi(x)$ is $cdf$ for standard normal distribution which means $$ ...
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1answer
40 views

Calculating probabilities for complex random variables

I am having some trouble understanding/formulating how one computes probabilites given a (somehow complex) continuous random variable. For example, if I define a random variable $Z$ as: ...
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26 views

The X & Y coordinates for points on a bell curve / normal distribution?

In Short: I want to give a formula the X coordinate and get the Y coordinate from matching a bell curve. Is this possible? In Detail: I'm trying to program a market simulation and to get a product's ...
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2answers
48 views

How to prove $E[e^{e^y}]=\infty$? y is a normal random variable

The question is, given $Y\sim N(\mu,\sigma^2)$, how to prove$E[e^{e^Y}]=\infty$? I tried to look Y as some kind of Ito's process and apply Ito's formula to it but it doesn't make sense. Next I tried ...
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1answer
31 views

Almost sure convergence of a sequence of Gaussians with vanishing variance

Let $(X_n)_{n\geq 1} $ a sequence of independent random variables. We assume that $X_n \sim \mathcal{N}(0,\sigma_n^2)$ and that $(\sigma_n)_{n\geq 1}$ is a vanishing sequence of positive numbers. Let ...
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1answer
33 views

Normal distribution with dice

I'm wondering how to control the normal distribution that comes from summing dice rolls only using different numbers of dice, different combination of types of dice (d4, d6, d8, d10, d12, d20) and ...
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1answer
22 views

Expectation of product of two correlated gaussian variables

$\newcommand{\var}{\operatorname{var}}$It seems I can not find the answer anywhere, please point it out how to calculate. Here, I have $X$, $Y$,$G$,$X_D$ and $Y_D$,both are Gaussian variables, and ...
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1answer
34 views

Variance of a Gaussian Random Variable

Show Variance of a Gaussian random variable $N(\mu,\sigma^2)$ and I know $\mathbb{E}(X)^2 = \mu^2$. So I need $\mathbb{E}(X^2)$ = $\int_{\mathbb{R}} x^2 \frac{1}{\sqrt{2\pi\sigma^2}} ...
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1answer
30 views

Sigmoid function that approaches infinity as x approaches infinity.

The function I'm looking for looks like an error function, but instead of having asymptotes $1$ and $-1$, the function I'm looking for does not have asymptote. It increases to infinity. The ...
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12 views

Regarding the distribution of pivotal functions not depending on their parameter(s)

I have difficulties understanding the part of pivotal functions not depending on their underlying parameters. Let's take a simple example, if Y is a random sample from an $N(\mu,1)$ distribution and ...
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44 views

Could we define two random variables such that the product of them is Normal distribution(Gaussian)?

Could we find two random variables $X$ and $Y$ which $XY \sim N(\mu, \sigma^2)$? I found the ratio of two normal distributed random variables is distributed Cauchy distribution. However, on the ...
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1answer
22 views

finding variance of gaussian distribution from mean

The Gaussian random variable $X$ can be used to model the number of customers that enter a market in 1 minute at a given time of the day. The mean number of customers that enter the market in 1 minute ...