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

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11
votes
1answer
74 views

Distribution of $\sum\limits_{i=1}^{N}X_{i}$ conditionally on $\sum\limits_{i=1}^{N}X_{i}^{2}$ for i.i.d. standard normal $X_i$s

Assume that the random variables $X_{i}$ are i.i.d $\mathcal{N}\left(0,1\right)$, then: $$S_N=\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)\qquad\qquad ...
1
vote
1answer
14 views

Random variable with density function that is scaled geometric mean of density functions of two independent normally distributed random variables

Given two independent normally distributed random variables A and B: $$A \sim \mathcal{N(\mu_A, \Sigma_A)}$$$$B \sim \mathcal{N(\mu_B, \Sigma_B)}$$ is there a way to find normally distributed random ...
0
votes
1answer
11 views

Modulus of Z (Normal distribution)

The random variable $Z$ is distributed such that $Z \sim N(0,1)$ find the probability of $P(\left|Z\right| >2.4)$. How to solve this modulus type of question ?
2
votes
1answer
22 views

Integrating a cost function over a normal distribution

Let's say you have a cost function $C(x)$ and you want to understand the expected cost if the input follows the normal distribution $$X \sim \mathcal{N}(\mu,\sigma ^2)\\ $$ If I want to find my ...
2
votes
2answers
560 views

Generalized chi distribution

Let $v\in\mathbb{R}^n$ follow a multivariate Gaussian$(0,I)$ distribution, and $M\in\mathbb{R}^{n\times n}$ a matrix. Has the distribution of the Euclidean norm $\|Mv\|$ been studied? I know that its ...
0
votes
3answers
47 views

Find the expectation of $Z=min\{X,Y\}$ which $X~N(\mu,{\sigma}^2)$,$Y~N(\mu,{\sigma}^2)$

Find the expectation of $Z=min\{X,Y\}$ which $X\sim N(\mu,{\sigma}^2)$,$Y\sim N(\mu,{\sigma}^2)$. $X$ and $Y$ are independent random variables. This is how far I go: According to order statistics, I ...
0
votes
1answer
18 views

Finding a point on a normal distribution knowing just two other points on it?

So I have two known points on a normal distribution. A 50th percentile point and a 75th percentile point. How can I figure out what percentile a third point is at? For instance: 50th percentile: ...
1
vote
1answer
28 views

Finding Marginal Density functions with $Y\sim N_4(\mu,\Sigma)$

Suppose $Y$ is $N_4(\mu, \Sigma)$ where $$\mu = ( 1,2,3,-2)'$$ and $$\Sigma =\begin{bmatrix} 4& 2& -1& 2 \\ 2& 6& 3& -2 \\ -1& 3& 5& -4 \\ 2& ...
-1
votes
0answers
10 views

Normal Distribution question involving stocks

An investor considers two stocks A and B. It is assumed that the return XA and XB for the two stocks over the next year are indepedent with XA ∼ N(0.175, 0.258) and XB ∼ N(0.055, 0.115). The return on ...
2
votes
1answer
28 views

Cholesky decomposition and variance

Well, I've been reading about simulating correlated data and I've come across Cholesky decomposition. Everything seemed clear until I found a couple of posts on this site and Cross-Validated that ...
0
votes
0answers
31 views

normal distribution under special condition

Given independent Gaussian random variables $U\sim N(−1,1)$ and $V\sim N(1,1)$, are the 2-element vector $T=(U+V, U−2V)$ and the variable $$W= U\text{ with 50% chance}, V \text{ with 50% ...
0
votes
2answers
37 views

How does the formula for standard deviation result in the normal distribution

Trying to understand this is in a high school level. I understand that the how $\frac {\Sigma|x-\bar x|}{n}$ calculates the mean of the distances of each score to the mean. I use this idea to map ...
0
votes
1answer
10 views

Parameters of gaussian distribution, which is generated using central limit theorem

In a software I am working on (sensor simulation), I needed to generate normally distributed noise for simulated sensor signals. I used the central limit theorem. I generated 20 random numbers and ...
0
votes
1answer
344 views

Gaussian distribution and its parameters

I need to learn more about Gaussian distribution and given a set of data, plot a Gaussian distribution of it. Using the following code sample, could you please tell me how I can plot a Gaussian ...
0
votes
2answers
16 views

Normal distribution with standard deviation = I

Suppose a vector $\epsilon \in \mathbb R^d$ is a random vector drawn from the isotropic normal distribution: $\epsilon$ ~ $\mathcal N (0, I)$ [As in Eq. 1.34 here.] I suppose ...
0
votes
0answers
7 views

How to calculate distribution of (X1, X2) conditional on (C1, C2)?

Say that $X_{1}$ = $a_{1}$$X_{2}$ + $B_{1}$$C_{1}$ + $E_{1}$  , and         $X_{2}$ = $a_{2}$$X_{1}$ + $B_{2}$$C_{2}$ + $E_{2}$  , ...
0
votes
0answers
11 views

SSE distribution in simple linear regression

I'm looking at the typical simple linear regression model $Y_i = \beta_0 + \beta_1X_i + \epsilon_i$, where there $\epsilon_i$s are iid $N(0, \sigma^2)$ random variables. We have unbiased estimates ...
1
vote
2answers
311 views

Assumption of a Random error term in a regression

In one of my recent statistics courses, our teacher introduced the linear regression model. The typical $y=\alpha + \beta X + \epsilon$, where $\epsilon$ is a "random" error term. The teacher then ...
0
votes
0answers
29 views

Solving for the limit of a Gaussian random variable within an integral

I'm having trouble solving a particular integral. It is $$ (1/\Delta t)\int_t^{t+\Delta t}I(t')dt', $$ where $$ I(t') = \mu_c+\sigma_c \eta(t'). $$ In this second equation, $$ \eta(t') = ...
0
votes
0answers
31 views

Distribution or samples of a function of a random variable

OK I edited the question: I have the following setup: Stereo camera setup with two images I, I'. 4 1-dimensional random variables (each corresponding to the inverse depth value of a pixel on an ...
0
votes
1answer
399 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
1
vote
1answer
335 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
0
votes
2answers
24 views

Generating points from 2 Normal distributions and $0$-probability continuous r.v.s

Consider the following experiment: We generate "green" points and "blue" points in $\mathbf{R}$ using two different normal distributions as follows: 1000 green points are sampled from a $N(-1, 1)$ ...
1
vote
2answers
25 views

Using Normal Distributions to find Proportion

The height of a randomly selected woman from a population is normal with $\mu=165cm$ and $\sigma=7cm$. The heights f the men in this population are normal with $\mu=178cm$ and $\sigma = 8cm$. I am ...
1
vote
1answer
419 views

Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we ...
-5
votes
0answers
23 views

Expected lifetime [closed]

I have 5 machines (elevators) and would like to know "a realistic worst case" in terms of when they break down and need to be replaced (for budgetting). They run independently and so far for 62 years ...
2
votes
1answer
31 views

Understanding the matrix normal distribution

A random $n \times p$ matrix $X$ is distributed according to a matrix valued normal distribution iff $\mathrm{vec}(X) \sim \mathcal{N}_{np}(\mu, V \otimes U)$, where $\mu \in \mathbb{R}^{np}$ is a ...
2
votes
1answer
47 views

Gaussian function in the limit of trigonometric functions

I've noticed that $$ (\sin\theta \cos\phi)^{2n} + (\sin\theta \cos\phi)^{2n-1} $$ increasingly resembles a Gaussian function of $(\theta, \phi)$ as $n$ goes to infinity. In particular, when I take ...
1
vote
1answer
454 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
1
vote
1answer
128 views

Finding 'symmetrical range' from mean.

A machine used to make butter where its masses are normally distributed with mean m and standard deviation s.It is found that 5% from these butters are having mass more than 85g where else 10% are of ...
0
votes
1answer
40 views

If $X$ and $Y$ are Normally distributed with correlation $\rho$, can we say anything about $E[Y \mid X]?$

Let $X \sim N(0, 1)$ and $Y \sim N(0, 1)$ and $\mathbb E[XY]=\rho$. Can one say anything about the conditional expectation $\mathbb E[X \mid Y]$? In general, this clearly does not seem to work, ...
3
votes
1answer
470 views

How to plot standard deviation rings of a bivariate normal distribution?

I'm working on a project right now where I have Gaussian distributions, and I want to create a graphic that represents them. I'm not sure how to generate the ellipse that represents say 1 standard ...
-1
votes
1answer
28 views

expectation of product of sums of normally distr. r.v.

Let $Z_1$ and $Z_2$ be i.i.d. standard normally distributed. $X_1=Z_1+Z_2$ and $X_2=Z_1-Z_2$. Apparantly E[|$X_1|*|X_2|$] = E$[|Z_1|*|Z_2|]$. Why?
1
vote
1answer
3k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
6
votes
0answers
61 views

Justify an unbiased estimator is UMVUE

Suppose $X_1,\ldots,X_n$ are iid $N(\theta,\theta)$, with $\theta\in(0,\infty)$. Is $\bar{X}$ the UMVUE (beta unbiased estimator) of $\theta$? I find the complete sufficient statistic is ...
2
votes
4answers
2k views

Scaling the normal distribution?

I might just be slow (or too drunk), but I'm seeing a conflict in the equations for adding two normals and scaling a normal. According to page 2 of this, if $X_1 \sim N(\mu_1,\sigma_1^2)$ and $X_2 ...
0
votes
1answer
28 views

What is the PDF, CDF, and E[Y] of Y=ln[X+c] if X is lognormal

If $\ln X \sim N(\mu, \sigma^2)$, what is the distribution of $Y=\ln \left(X+c\right)$ where $c$ is a constant. Is this something that can be written out analytically? Also, what is $E[Y]$?
3
votes
1answer
226 views

Distribution of higher powers than 2 of a gaussian distribution

If $X \sim \mathcal{N}(0,1)$, then $X^2 \sim \chi^2(1)$. What about higher powers of $X$? I know that the Gamma Distribution is a generalization of the $\chi^2$ distribution, but I don't know how the ...
0
votes
2answers
392 views

normal distribution using Z - finding probability between 2 numbers

I am wanting to find the probability of the following: SD = 20 Mean = 100 P(85 < X < 117) i have found the z values for both: P(X>85) : X-u/o = 85-100/20 Z = -0.75 and found the ...
1
vote
2answers
23 views

Probability more than 25% greater?

The random variable X is distributed N(60,64). The random variable Y is distributed N(52,36). Find the probability that a random observation from X is more than 25% greater than a random observation ...
1
vote
1answer
315 views

Getting a Hermite polynomial expansion of Gaussian with given variance.

I am trying to find an expansion of centered Gaussian - $\frac{1}{\sqrt{2\pi}\sigma}\exp({-\frac{x^2}{2\sigma^2})}$ in terms of Hermite polynomials. Namely to calculate ...
1
vote
2answers
31 views

Estimate mean and variance for a truncated sample set

Assume there is a normally distributed random variable $X \tilde{} N(\mu, \sigma)$ I want to estimate $\mu$ and $\sigma$. So far the standard setting. Assume I am given a sample $(X_i)_{i=1}^N$ of ...
0
votes
1answer
30 views

What is the correct equation for “Normal distribution function of continuous random variable”?

I was reading a book and came across with a equation which gives the normal distribution function of continuous random variable. It was used in a software called ...
2
votes
1answer
88 views

Minimum matching convolution

Let $\text{SPD}^n$ and $\text{PD}^n$ be the symmetric semi-positive and positive definite matrices in $\mathbb{R}^{n\times n}$, respectively. I want to find an $X\in \textrm{SPD}^n$ that minimizes ...
1
vote
1answer
12 views

T distribution problem

I will be using $t$-distribution to solve this problem. Specifically,the pooled variance test because both samples have size less than $30$,and both populations seem to have the same population ...
1
vote
1answer
59 views

Standard normal distribution hazard rate

Is the hazard rate of the standard normal distribution convex? Can you give a reference?
1
vote
0answers
50 views

Integrating a prob distr over the set of possible circles within an annulus

Let $z$ be the measured coordinates of a point on a circle $c$ with center $x$ and radius $r$. Assume the probability of measuring $z$ given the circle $c$ is normally distributed by the distance ...
0
votes
2answers
67 views

Covariance and normal distribution

Let $X,Y ∼ N(0,1)$ be i.i.d. and let $U,V$ given by $U=aX+bY+c$ and $V=dX+eY+f$ have a bivariate normal distribution (here $a, b, c, d, e, f ∈ R$ with $ae − bd$ not equals to 0). (a) What is $Cov(X, ...
1
vote
2answers
27 views

What's $r$ going to be when you get the summation of $36$ Geometric $X_i$'s

Let $X_1,X_2,\ldots,X_{36}$ be a random sample of size $n=36$ from the geometric distribution with the p.d.f: $$f(x) = \left(\frac{1}{4}\right)^{x-1} \left(\frac{3}{4}\right), x = 0,1,2,\ldots$$ Now ...