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

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0
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1answer
23 views

How to find the subspace on which a multivariate normal distribution is concentrated?

Let $(X_1,X_2,X_3)^T$ be a multivariate normal distribution which is singular (that is, its variance matrix has determinant $0$). How can we find the subspace $U \subset \mathbb{R}^3$ with dimension ...
0
votes
1answer
25 views

Question about three Normal distributions

Let there be 3 normal distributions $X\sim\mathcal{N}(\mu_x,\sigma_x)$, $Y\sim\mathcal{N}(\mu_y,\sigma_y)$, $Z\sim\mathcal{N}(\mu_z,\sigma_z)$ and 3 random samples from each distribution- $x,y,z$. ...
0
votes
1answer
42 views

Is $E(\sin Z)=\sin E(Z)$ for $Z\sim N(0,1)$?

Let $Z$ be a random variable of the standarized normal distribution. True or false? $E(\sin Z)=\sin (E(Z))$ ($E$ stands for the mean value). Comments. Since $\displaystyle \sin ...
4
votes
2answers
207 views

Probability of Normal Distribution with Unknown Mean

I am still quite new to the whole idea of probability and statistics and am not sure how to do this question. The random variable R, also normally distributed, has a standard deviation of 3.59 with ...
1
vote
1answer
23 views

Integrating inverse cumulative of standard Normal Distribution

I am studying this book and a particular line reads $\int_{1-p}^1 \Phi^{-1}(u)du=${set $k=\Phi^{-1}(u)$}=$\int_{\Phi^{-1}(1-p)}^\infty k\phi(k)du$, where $\Phi$ and $\phi$ are the cumulative and ...
1
vote
1answer
34 views

creating positive definite matrix using wishrnd in matlab

I used the below code to create a matrix of wishart distribution, ...
0
votes
1answer
14 views

Modelling a Bivariate Normal Distribution in Matlab

Hi I was wondering would anyone have experience with modelling multiple bi-variate Gaussian distributions on the same plane in Matlab? Say we know the coordinates of the means, let m1 = mean1= [1 2] ...
0
votes
0answers
25 views

How to calculate sample variance?

I have an infinite population. Out of this population, I'm picking $N=40$ items, each of which can have a value of $0$ or $1$. Say I get a mean of $0.7$. How do I calculate the standard error of this ...
0
votes
1answer
23 views

Recover Marginal Distribution subject to a Constraint

I want to identify the marginal of a normal distribution subject to a restriction. Take two normally distributed random variables $x,y$. Their pdfs are denoted by$\phi(x)$ and $\phi(y)$. The moments ...
2
votes
1answer
39 views

Exponential law with both positive and negative values

The exponential law with density $f(x) = \lambda e^{-\lambda x}$ for $x \geq 0$ and $f(x)=0$ for $x < 0$, is well-known. What's the name of the distribution which has $$f(x) = \frac{1}{2} ...
1
vote
1answer
25 views

on a quantisation of the bell curve

The bell curve function: $e^{-x^2/2}$ is an eigenfunction of the Fourier transform (FT) on the real line. Is its quantisation/discretisation the binomial distribution (coefficients $n$ choose $k$) an ...
1
vote
1answer
67 views

Multivariate normal value standardization

I am wonder how to standardize multivariate normal value. Normal standard multivariate distribution of $q$ variables is $z\sim N_q(0_q,I_q)$. I have found that $Bx\sim N_q(Ba,B\Sigma B^T)$ and based ...
2
votes
1answer
56 views

Maximizing the probability of a poll prediction

Using the central limit theorem, I was able to find out the first part of this question. However, part b is eluding me. How do I, in general, find a value for $n$ such that we can ensure the ...
-1
votes
2answers
35 views

What random variable is this?

I have a sequence of reals $S = s_1,s_2,\dots,s_n$ such that $s_i-s_{i-1}$ is a Gaussian distribution. From histogram of sequence $S$ (10000 elements) it appears that it is uniform distribution. Is it ...
2
votes
0answers
38 views

Why is Gaussian matrix full rank?

Suppose $A\in R^{n\times n}$ is a matrix with independent standard normal entries. Is there an elementary argument to show that $A$ is nonsingular with probability $1$?
1
vote
1answer
22 views

Independence: norm v.s. direction of a standard multivariate normal vector

Suppose that $v\sim N(0,\sigma^2 I_n)$ and with $||\cdot||$ denoting the Euclidean norm, define $$ u=v/||v||\quad\text{and}\quad w=||v||. $$ I've been told that $u$ and $w$ are independent and I see ...
1
vote
2answers
45 views

Finding percentile given distance between two percentiles.

The sales for a company are normally distributed with mean $\mu$ and variance $\sigma^2$. The difference between the $90$th and $40$th percentile is $500$. The $70$th percentile is $1700$. What is the ...
2
votes
1answer
36 views

Expectation of scaled sum of squares of iid random variables

Let $X_1, \dots, X_n$ be iid standard normal random variables. Consider the vector $X = (X_1, \dots, X_n)$ and the vector $Y = \frac{1}{\|X\|}(X_1, \dots, X_k)$ for $k < n$. What is ...
0
votes
0answers
12 views

The random graph quantity $S(n: K, L)$

I am going through Degree sequences of random graphs by Béla Bollobás. On page $3$ the author introduces the quantity $S(n: K, L)$ without any explanation. Could anyone please help me in ...
0
votes
1answer
36 views

IMPROVED - Proving that a statistics is not sufficient (Gaussian case).

Let $X=(X_1,...,X_n)$ be i.i.d. $N(0,\sigma^2)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic? I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient ...
0
votes
0answers
12 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
24 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
52 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
38 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
11 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
30 views

Separability Hypothesis Test

Let $M_1 \sim \mathcal{N}(\mu_1,\mu_1)$ and $M_2 \sim \mathcal{N}(\mu_2,\mu_2)$ be normally distributed and independent random variables, each depending on only one parameter $\mu_n$ ($n = 1,2$). ...
0
votes
0answers
23 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
25 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
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0answers
27 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
54 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
40 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
365 views

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
1answer
14 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
46 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
30 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
33 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
35 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
48 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
19 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
33 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
23 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
34 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 ...
2
votes
1answer
53 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
33 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
26 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
70 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
17 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
22 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, ...
2
votes
0answers
37 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
23 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 ...