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

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-2
votes
0answers
14 views

Distribution of a sum of squared normal variables? [on hold]

What is the distribution of $\sum X_i^2$ ? Where $X_i\sim N(\mu_i,\sigma_i^2)$, for $i=1,..,n$. Thanks a lot for answers!
0
votes
0answers
6 views

Optimize polling frequency between producer and consumer to achieve minimum waiting time

Background: I am trying to optimize what we call AJAX request polling frequency in the domain of web design, and I wanted to check if I could use some help from math guys to explore a better ...
1
vote
1answer
15 views

Method for determining distributions of sum of Normal distribution unknown mean and variance

I've been trying to complete this question but have been struggling to see how to approach it. Any help would be greatly appreciated. Is there a standard way of approaching and answering ...
2
votes
0answers
47 views

Expectation and Variance of $X/(X+Y+Z)$

I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius.. $X \sim \mathcal N(\mu_1,\sigma_1)$, $Y \sim \mathcal N(\mu_2,\sigma_2)$, $Z \sim \mathcal ...
2
votes
1answer
14 views

Inverse of Gaussian CDF, Sum

Consider the following setting. Let $k = 1, \ldots, n$ and define $$y_k= \Phi^{-1}\left(\frac{k}{n+1}\right),$$ where $\Phi$ is the inverse of the CDF of a standard normal. I noticed numerically ...
0
votes
0answers
6 views

Bounding the ratio of sample covariance to population covariance

I am looking to bound the Kullback Leibler divergence of two mean centered Gaussian laws $\mathbb{P}_1=\mathcal{N}(0, \Sigma)$ and $\mathbb{P}_2=\mathcal{N}(0, \hat{\Sigma})$ where $\Sigma$ is the ...
0
votes
0answers
10 views

Separability Hypothesis Test

Let $M_1 \sim \mathcal{N}(\mu_1,\mu_1)$ and $M_2 \sim \mathcal(\mu_2,\mu_2)$ be normally distributed and independent random variables, each depending on only one parameter $\mu_n$ ($n = 1,2$). Let's ...
0
votes
0answers
14 views

integral of trace function

How can I compute the below integral? $$ \int e^{-1/2*tr[(\mu\mu^T-\mu m^T-m\mu^T)\Sigma]}d\mu $$ in which $\mu \in R^{n},m \in R^{n},\Sigma \in R^{n*n}$ and $tr(.)$ is trace of matrix. I have ...
1
vote
0answers
21 views

Integer and fractional pars of Gaussian random variables

Are there any interesting results known about integer and fractional parts of $\xi \in \mathcal N(0, \sigma^2)$? In particular, I am interested in their expected values and covariance matrix.
1
vote
0answers
20 views

Joint distribution of two normal marginal distributions

My question is related to the possibility of stating joint convergence in distribution from marginal weak convergence. Consider two sequences of random vectors $X_n$ and $Y_n$ defined on the ...
2
votes
0answers
49 views

Prove that the limit in probability of normally distributed random variables is normally distributed, too [duplicate]

Let $X_n\sim\mathcal N_{\mu_n,\sigma_n^2}$ for some $(\mu_n,\sigma_n^2)\in\mathbb R\times(0,\infty)$ and $X$ be a real-valued random variable with $$X_n\stackrel{\text{in probability}}\to X\;.\tag 1$$ ...
1
vote
1answer
35 views

Joint characteristic function of $x$ and $y=x^2$ if $x$ is the standard normal variable

How to find the joint characteristic function of $x$ and $y=x^2$ if $x$ is standard normal variable with mean $0$ and variance $1$?
8
votes
3answers
299 views
+50

Maximum of a sum of random variables

Let $X_1, \dots, X_n$ be independent and identically distributed random variables with $E(X_i) = 0$ and $$S_k = \sum_{i \leq k} X_i$$ What is the probability distribution of $M_2 = \max \{ X_1, ...
0
votes
0answers
10 views

Estimation effectiveness of two normal-distributed variables

You have two processes of measuring the air pollution, $X$ and $Y$. Both processes deliver values which are normal distributed around $\mu$: $X ~ N(\mu, \sigma_x^2)$ and $Y ~ N(\mu, \sigma_y^2)$. I ...
1
vote
0answers
40 views

Distribution of Product of Matrices holding Multivariate Random Normal Variable Observations

We have matrices $\boldsymbol{A}$ and $\boldsymbol{B}$ that hold observations from multivariate normal distributions. Each matrix represents $\boldsymbol{n}$ observations of $\boldsymbol{m}$ variables ...
0
votes
1answer
24 views

Understanding standardization for normal distribution

Let X be normally distributed random variable with expected value $\mu$ and standard deviation $\sigma$, then its СDF is: $$ F(x)=\frac{1}{\sigma\sqrt{2\pi}} \int_{-\infty}^x ...
1
vote
2answers
28 views

$\frac{1}{\sqrt{2\pi}}\int_a^\infty e^{-\frac{1}{2}(x-\mu)^2}dx$ - Normal Distribuition

I have read in one of my finance books (Asset Pricing - John H. Cochrane) that there is this identity: \begin{equation} \begin{split} \frac{1}{\sqrt{2\pi}}\int_a^\infty e^{-\frac{1}{2}(x-\mu)^2}dx ...
0
votes
1answer
25 views

Finding the conditional distribution of 2 dependent normal random variables

Here's the situation $X \sim N(\mu, \sigma^2)$ and given $X=x$, $Y \sim N(x, \tau^2)$ I need to find the distribution of $X$ given $Y=y$ From what's given, I know the pdf's of $X$ as well as ...
0
votes
1answer
33 views

Black-Scholes SDE solution help

I am trying to solve the Black Scholes SDE, but got really stuck. I have done most of the derivation but the integral seem intractable to me. The integrals look bit like the Normal Distributions PDF, ...
0
votes
0answers
18 views

When studying 2D gabor functions why is a gaussian called elliptical?

Consider $$G(x,y)=\frac{1}{2\pi\sigma\beta}e^{-\pi\left[\frac{(x-x_0)^2}{\sigma^2}+\frac{(y-y_0)^2}{\beta^2}\right]}e^{i[\xi_0x+\nu_0y]}.$$ This is the product of a complex plane wave and what this ...
2
votes
2answers
19 views

zero covariance but not independent - normally distributed random variable $X$ and $X^2$

This is one of my homework question, which the answer sheet has already been given out. However, I still don't understand it. Exercise 1.1. It is well known that for two normal random variables, zero ...
0
votes
0answers
21 views

Expected value log-normal variable

Suppose $X_t$ and $Y_t$ both have standard normal distributions with mean zero and variance 1. What is the $E_t[e^{aX_t + bY_t^2}]$ where $a$ and $b$ are constants? It should be something like: ...
-1
votes
1answer
31 views

Prove inequality for tail of normal distribution

I have to prove this inequality for $x>0$. I have no idea how to even start this. I would appreciate any help. $$\frac{x^{2}}{1+x^{2}}\frac{1}{x}\exp \left( \frac{-x^{2}}{2}\right)\leqslant ...
-1
votes
0answers
9 views

Application of Jacobian matrix determinant in multivariate normal distribution transformation

Consider a multivariate normal distribution in variable x with mean μ and covariance Σ. Show that if we make the linear transformation $y = Ax + b$ then the transformed variable y is distributed ...
2
votes
1answer
47 views

Expectation of a function of a normally distributed random variable

Consider that I have to produce this result: $$E[u(W_0+r(\theta))] = u(W_0)+\theta-\frac 12\rho\sigma^2$$ From this: $$ E[u(W_0+r(\theta))] = \int_{-\infty}^\infty u(w_0+r) \frac{1}{\sigma ...
1
vote
2answers
25 views

Confidence Intervals

Let $x_1, \ldots, x_n$ be a sample from a normal population having unknown mean and variance. Let $\bar{x}$ be the average of the first n of them. What is the distribution of $x_{n+1} - \bar{x}$? If ...
0
votes
2answers
20 views

convergence in distribution of truncated gaussian variables

Let $X$ be a random variable which is distributed normally with mean $\mu=0$ and variance $\sigma=1$. Suppose that $X_n$ is a random variable for any positive integer $n$ with truncated normal ...
0
votes
1answer
21 views

Find the value for z(0.1) from a distribution table?

I'm doing a statistics course, and I thought I had no problems using distribution tables to find values. For example, for the Gauss distribution, if I want $\Phi(-2)$, I will do $1 - \Phi(2)$ because ...
0
votes
0answers
14 views

segments of normal distribution are normally distributed?

I need a hint how to prove following: Log fold changes follow normal distribution. On the plot you can see log2 fold changes versus mean. If I segment log2 fold changes into the bins, so that I have ...
0
votes
1answer
18 views

How to find the PDF for Y=|X-1|?

Given $X\sim N(0,1)$ and $Y=|X-1|$. Find the PDF of $Y$. I tried to discussed when $x>1$ and $x\le1$, but this gives me two different functions and I have no idea how to combine them. However, ...
1
vote
0answers
17 views

Expectation of the matrix product $Q A (Q A)^h$

Let $Q$, of dimension $m \times n$, be a matrix with i.i.d. complex Gaussian elements; we suppose that these elements have zero mean and have the same variance $= v$. We define $A$ as an $n \times k$ ...
0
votes
0answers
20 views

Distribution of discrete function of continuous random variable?

It has been quite some time that I did statistics, and I am not sure how to figure out the distribution of a function of a random variable if the function itself discretizes (if that is a word) the ...
1
vote
0answers
47 views

The higher moments of truncated Gaussian

We assume that $$X \sim N(0,1/d),$$ where $d\rightarrow \infty$. For $\delta > 0$ sufficiently small. My question is, what is the correct order (in terms of $d$) of $$ ...
0
votes
0answers
24 views

Could you please explain how do we get the second term while computing $E[f(Y)]$

I will fix the dimension $n$ and use $S:=\{x\in\Bbb R^n:\|x\|=1\}$ to denote the unit sphere. Let $\sigma$ denote surface measure on $S$, and define $\bar\sigma:=[\sigma(S)]^{-1}\sigma$, the "uniform ...
3
votes
0answers
55 views

Compound Distribution — Log Normal Distribution with Log Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (actually, the mean ...
2
votes
0answers
35 views

Compound Distribution — Log Normal Distribution with Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Log Normal Distribution whose mean (mean of the log of ...
2
votes
1answer
42 views

Compound Distribution — Normal Distribution with Log Normally Distributed Variance

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Normal Distribution whose variance is distributed Log ...
3
votes
0answers
33 views

Compound Distribution — Normal Distribution with Normally Distributed Mean

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Normal Distribution whose mean is distributed Normally. ...
1
vote
0answers
6 views

Can you perform a two-sample t-test with a moderately skewed sample distribution

This was a question on my AP Stat test that I got wrong and I am not sure if it is correct: Essentially the question stated you have two samples both of which don't satisfy the large counts condition ...
0
votes
1answer
23 views

Integrate Gaussian function

I am trying to integrate Gaussian distribution from -m to m to find parametar A. I have done this so far: $\int_{-m}^{m}\frac{A}{\sqrt(2\pi)\sigma}e^{-(x-m)^{2}/(2\sigma^{2})}dx=1$ after ...
2
votes
0answers
46 views

Probabilty concerning two normal distributed random variables

Consider the two independent random variables $X$ and $Y$ where each random variable is 3-dimensionally normal distributed with $X \sim \mathcal{N}(\mathbf{0},\Sigma_X)$ and $Y \sim ...
0
votes
0answers
25 views

Can we illustrate pdf of Normal distribution with mean and variance following Normal by analytic expression?

I just wonder if pdf of Normal distribution with mean and variance which are normally distributed can be expressed in analytic formula, i.e. $ \mathcal{N}(\mu, \sigma^2) $ where $ \mu $ ~ $ ...
0
votes
0answers
10 views

creating matrix normal distribution

would you please help me? I have a distribution on $vec(A)$ as below $$ q(vec(A))=N_{np}(A|vec(\mu),Q) $$ In which $N(.)$ means the normal distribution and $$ Q=L\otimes K+ H\otimes J $$ How can ...
0
votes
0answers
25 views

Test for equality of two multivariate normal distributions for given µ and sigma

As title states, I'm interested in finding a proper inferential test to check if two multivariate normal distributions with given mean and standard deviation vectors have been draw from a common ...
0
votes
0answers
14 views

Variational inference on a Normal distribution: is my choice of priors passable?

I am trying to understand the basics of Variational Inference. In order to do so I designed a very simple problem: using the free-form mean field method to approximate the posteriori distribution of ...
0
votes
0answers
10 views

Correlate normal shocks

I am trying to generate some random standard normal variables and correlate them In particular I want: $$ \bf Y \sim \mathcal N(0, \Sigma) $$ where $\textbf{Y} = (Y_1,\dots,Y_n)$ is the vector I ...
0
votes
1answer
45 views

Variance of squared random variable

Can anyone help to prove this equation for any distribution $$ E(z^4)=1+\operatorname{Var}(z^2) $$ where $z$ is a random variable with the standard normal distribution $$z=\frac{x−μ}σ$$
0
votes
1answer
15 views

Unconditional Variance of Normal RV with mean being a NRV

I am trying to find the variance of $X$ which is defined like this: $$X \sim N(Y,e)$$ where $Y$ is a normal random variable with the distribution $Y \sim N(a,b)$. $a$,$b$, and $e$ are known ...
3
votes
2answers
39 views

Log - Normal Distribution

could someone explain why the log-normal distribution's mean is $$ e^{u + {\sigma^2\over2} } $$ and the variance is $$ (e^{\sigma^2} -1)e^{{2u} + \sigma^2} $$ I'm not too sure how ...
0
votes
0answers
14 views

Multivariate probit gaussian convolution

For univariate normal distribution, we know the following formula exists $\int\Phi(a)\mathcal{N}(a|\mu,\sigma^2)da=\Phi\left(\frac{\mu}{\sqrt{1+\sigma^2}}\right)$ Is there a similar formula for ...