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

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0
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0answers
6 views

Find expected step, on which half-normal RV exceeds a scalar value?

I have defined a following problem. Given is a non-negative integer variable (steps) $s\in[0,1,...)$, and a scalar random variable as a function of $s$, $R(s)$. Random variable is half-normally ...
-2
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1answer
20 views

Finding distribution of $Y$ and $P(Y>1)$

$X$ is a random variable which has normal distribution with mean $4$, variance $9$ and $Y=3X-8$. What is the distribution of the random variable $Y$? How should I find $P(Y>1)$?
1
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1answer
23 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
31 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
14 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
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2answers
34 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
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0answers
18 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
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1answer
15 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 ...
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2answers
30 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 ...
1
vote
1answer
21 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 ...
-1
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0answers
32 views

a) Five men selected at random from a normal population with mean weight p= [on hold]

a) Five men selected at random from a normal population with mean weight p=160 lb and o- = 2 .00 lb, get on elevator. What is the probability that: i) All five men weigh more than 170? ii) The ...
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0answers
10 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 ...
1
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1answer
30 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 ...
-2
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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
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0answers
8 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
17 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
48 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
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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
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0answers
11 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
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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
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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.
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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
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0answers
50 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 $$X_n\stackrel{\text{in probability}}\to X\;.\tag 1$$ ...
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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
306 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, ...
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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
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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
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2answers
31 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
28 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
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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, ...
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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
20 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
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0answers
22 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
11 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
21 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
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1answer
22 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
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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, ...
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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 $n \times k$ ...
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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
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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
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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
57 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
36 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 ...