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

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1answer
33 views

Complex circular symmetric Gaussian and real Gaussian

Circular symmetric complex Gaussian zero mean PDF is defined as : $$f(z)= \frac{1}{\pi^N||M||} e^{-z^*M^{-1}z} $$ where $M$ is hermitian semi positive definite, $z \in \mathbb{C}^{N \times 1}$ and ...
1
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1answer
29 views

Square of Normal distributed variable?

This is a quick question. If $X\sim\operatorname{Normal}$, is $X^2$ rayleigh distributed? I ask this question is because from wiki, it says $X^2$ is called the Chi-Squared Distribution with a degree ...
1
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0answers
46 views

Random numbers generator

If I know how to generate random numbers from Gaussian distribution (using Box-Muller method), how can I generate random numbers from distribution with pdf ...
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1answer
19 views

Limiting Behaviour of Root Mean Square Normal Random Variables - Related to Chi-Squared Distribution

Above is my question. I have done the first part - made hard work of it, albeit, but still, it's done. The next part is where I am stuck. Intuitively, it seems (to me!) like we should have $R_n ...
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3answers
28 views
0
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0answers
23 views

Non-linear transformation of symmetric distribution to get non-negative skewness

Say you have a variable $x \sim D(\mu,\sigma^2) $, where $D$ is a symmetric known distribution. I'm looking for two linear or non-linear transformations of $x$ that give one negative and one positive ...
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0answers
29 views

When computing normal distribution values in applications, should one round initial value?

I know how to compute the basic normal distribution problems but I have a question on rounding. The problem is We have bags of candies with average weight 150 g and standard deviation 5 g. The ...
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0answers
5 views

Conjugate Gaussian Prior

Suppose we have a univariate Gaussian distribution as our likelihood, and now we have our prior belief to be multivariate Gaussian distribution on our parameters, so what will the posterior be? Is it ...
2
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1answer
29 views

Proof of mean and vector

Let $X_1,\ldots,X_n$ be a random sample from $N(\mu, \sigma^2)$. Show that the sample mean and each $X_i-\bar X, i= 1,\ldots,n$, are iid. Actually $\bar X$ and the vector $(X_1-\bar X,\ldots,X_n-\bar ...
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0answers
22 views

Integrating to find area under probability density function of skewed distribution.

From flawr's answer to this earlier question of mine you can see that the probability density function of a skewed distribution is: $$f_a(x) = \frac{\phi(x)\Phi(ax)}{\Phi(0)} \qquad (*)$$ where ...
0
votes
1answer
19 views

Probability inference of an action from a continuous outcome

Assume person A takes an action, it could be either $a_1$ or $a_2$ with $a_1>a_2$, we cannot observe A's action but a signal $x$, with $x=a_i+\epsilon$. $\epsilon$ follows a normal distribution ...
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0answers
15 views

Multivariate Normal Density Concavity

For this variance compunent model $Y$~$N(X\beta, \Omega)$, where $\Omega=\sum_{i=1}^m\sigma_i^2V_i$, the log likelihood function is $(\beta, \sigma_1^1, ..., \sigma_m^2)=C+\frac12\log ...
1
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0answers
33 views

Convergence in distribution of normal random variables

Let $X_n \sim \mathcal{N}(\mu_n,\sigma_n^2)$. Prove that if $X_n \rightarrow X$ in distribution, then either $X$ is normally distributed or there exists a constant $c$ such that $X = c$ almost surely. ...
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1answer
27 views

How do you prove 2 normal random variables X and Y are jointly normally distributed?

How do you prove 2 normal random variables X and Y are jointly normally distributed? I know that any linear sum of X and Y should be normally distributed but how do you prove that?
0
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1answer
23 views

Sample Size Estimation with unknown variance

After spending a lot of time I'm still nowhere. What I have: Let d denote the desired margin of error so that the interval for µ is of width 2d. Let (1 − α) denote the probability associated with ...
0
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0answers
27 views

Worst case for $n$ Poisson trials?

I have $n$ Poisson events which occur with parameter $\lambda$. What can I expect the lowest of these to be? I'd be happy with any reasonable interpretation of the question, including "what is the ...
0
votes
1answer
52 views

Why erf(a-b)+erf(a)+erf(a+b) is so close to 3erf(a)?

I am approximating an empirical distribution function with a sum of three gaussians, and noticed that for erf function $erf(a-b)+erf(a)+erf(a+b)$ is numerically very close to $3erf(a)$ for applicable ...
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2answers
35 views

Convolutions and the Gaussian distribution

Suppose $X_1$ and $X_2$ are independent random variables each with the standard Gaussian distribution. Compute, using convolutions, the density of the distribution of $X_1 + X_2$ and show $X_1 + X_2 = ...
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0answers
12 views

The variance of a multivariate normal random variable

Suppose $\vec{X}$ is an N-dimentional random vector that is multivariate normal distributed: $\vec{X} = [X_1, X_2, ..., X_n]^T$ and $X_i \sim N(0,s_i^2)$ and all correlations bewteen $X_i$ and $X_j$ ...
0
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0answers
27 views

Moment List for Standard Normal Distribution

I am stuck trying to find the moment list for a standard normal. I have been told I can find it the similar way for exponential distributions using taylor series. I know the MGF = e^((1/2)(t^2)) for ...
1
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2answers
56 views

Find a 95% confidence interval on a binomial process.

Let's say that $73\%$ of $1506$ people interviewed were in favor of legalizing gay marriage. What is the $95\%$ confidence interval for the percentage of the public that are in favor of legalizing gay ...
0
votes
1answer
28 views

Normal distribution

X follows a regular normal distribution on V with center $\xi$ and inner product $<\cdot,\cdot>$, and let $\eta \neq 0$ be a vector in V. Show that the reel stochastic variable ...
0
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0answers
33 views

What is the probability of two things happening at the same time?

I am using the normal distribution for two events so there is a 34% chance of each event having one standard deviation above the mean. What is the probability of both events having one standard ...
1
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1answer
57 views

Limit of Gaussian random variables is Gaussian?

Consider a sequence $X_n$ of Gaussian random variables with mean $\mu_n$ and variance $\sigma_n^2$, which converges in distribution (to some limiting distribution). Can I then conclude that $\mu_n$ ...
0
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1answer
87 views

Finding Moment Generating Function of Normal Distribution

I need to show that the moment generating function of $Y$ is $$M(t)=(1 − 2σ^{2}t)^{−1/2}$$ where $X$ ∼ $N$($0$, $σ^{2}$) and that $Y$ = $X^{2}$. I know the moment generating function of $Z$ is ...
0
votes
1answer
17 views

Covariance in normal lognormal (NLN) mixture

Let $u = \epsilon e^{\frac{1}{2} \eta}$ where \begin{equation*} \left( \begin{array}{c} \epsilon \\ \eta \\ \end{array} \right) \sim N\left( \left( \begin{array}{c} 0 \\ 0 \\ ...
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votes
2answers
25 views

Covariance of normally distributed random variables

If $ X \sim N(0,1) $ and given $ X = x $ then $ Y \sim N(x,1) $ I want to find the $ Cov(X,Y) $ using the relationship stated above. My attempt: $ Cov(X,Y) = E[XY] - E[X]E[Y] \\ E[X] = 0\\ ...
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0answers
34 views

Expected Z score for tail end of a standard normal distribution

Good day, For a Standard Normal Distribution (Mean: 0 and Variance: 1), I need to find the expected z-score for the section that makes up the final 2.5% (0.025) of the distribution. So, finding the ...
2
votes
1answer
31 views

Find marginal distribution of $Y$ where $Y\mid X$ is $N(a_1+a_2X,\sigma_1^2)$ and $X$ is $N(\mu,\sigma^2)$?

Let a random variable $X$ be normal $N(\mu,\sigma^2)$ and let the conditional distribution of $Y$ given $X$ be normal $N(a_1+a_2X,\sigma_1^2)$. a)Find the joint probability density function of $X$ ...
0
votes
1answer
26 views

Log-normal distribution

If $X \sim LogN(\mu,\sigma ^2) $, would the distribution for $aX \sim LogN(\mu+a,\sigma ^2) $ for $ a>0$? My solution: $log(X) \sim N(\mu,\sigma ^2) \\ log(aX) = log(a) + log(X) \\ log(aX) \sim ...
1
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0answers
30 views

A interesting question about moments.

Let $\{X_n\}$ be a random variable sequence and $X\sim N(0,\sigma)$. In general, the convergence $E(X_n^k) \stackrel{n}{\longrightarrow}E(X^k)$ doesn't implie that $E(X_n^{k+1}) ...
0
votes
1answer
60 views

Supremum of a sequence of i.i.d. standard normal random variables [closed]

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. standard normal distributed random variables. Show, using Borel-Cantelli's Lemma, that $$\sup_{n\in \mathbb{N}} |X_n|= \infty ...
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votes
2answers
45 views

The probability involving the ordered normal sample.

Let $X_1,X_2,X_3,X_4,X_5$ be a normal sample, taken from the distribution with unknown mean $\mu$ and known variance $\sigma^2$. Calculate $$P(|X_{3:5}-\mu|<0.841\cdot\sigma).$$ Comment. If the ...
1
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0answers
31 views

How to find the compound of poisson and normal distribution?

how to find the compound distribution, if the rate of poisson distribution is normally distributed with mean and variance ? I know I have to find the integral of: $$ \frac {1} {\sigma \sqrt{2 \pi} ...
1
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0answers
61 views

Uniform integrability of specific sequence of RV

I am investigating the following limit $$ \lim_{n \to \infty} E \left[ n \ln^-\left(1 - 2 \frac{\sigma}{n} [{\cal N}]_1 + \frac{\sigma^2}{n^2} \underbrace{ \| {\cal N} \|^2}_{\chi^2 \mbox{ ...
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0answers
21 views

Is a circular and spherical gaussian the same thing?

This pdf https://www.cs.princeton.edu/courses/archive/spring07/cos424/scribe_notes/0419.pdf mentions "circular gaussians" as the simplest gaussian in 2 dimensions, what you get if you have I as the ...
2
votes
2answers
92 views

Central Limit Theorem, why $n \ge 30$?

This is what I think the technical statement of CLT is: If we consider $\overline{X}_{n}$ coming from a sample of $\mathcal n$ independent and identically distributed random variables $X_{i}$ with ...
1
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1answer
22 views

Independence of two multivariate normals.

Suppose we have two multivariate normals $X_1 \sim N(u_1, \Sigma_{11}\Sigma_{22}$) and $X_2 \sim N(u_2, \Sigma_{21} \Sigma_{22})$ . Why are $X_2 $ and $X_1-\Sigma_{12} \Sigma_{22}^{-1}X_2$ ...
1
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1answer
44 views

Showing That Two Normal-Based Random Variables Have the Same Distribution

Above is my question. $\overline X$ has distribution $N(0,1/n)$ - that's fine to work out. Similarly, $X_n / \sqrt{n}$ has distribution $N(0,1/n)$. These follow from the general relation $$ ...
0
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0answers
15 views

On Conditional distribution of the multivariate normal.

Following the answer to this question. Where we are talking about a multivariate normal than has mean and covariance matrix that can be decomposed as: $\boldsymbol\mu = \begin{bmatrix} ...
0
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0answers
19 views

Calculation of probabilities in Z table

I would like to calculate at least one probability from z table. I know that pdf for N(0,1) is 1/(2*pi)*exp^(-(x^2/2)). Also the cdf is However, I do not know how to calculate this integral. ...
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0answers
18 views

Approximate a function with a gaussian distribution.

I have a function which has a bell-type graph and i need to find a Gaussian(Normal) with the appropriate mean, variance and constant factor which is close to the original function.The function in ...
0
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0answers
23 views

Numerically stable way to compute the conditional covariance matrix

The Wikipedia article on multivariate normal distribution contains the well-known fact about the conditional "sub-distribution": If $μ$ and $Σ$ are partitioned as follows: $$ \boldsymbol\mu = ...
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0answers
44 views

Normal distribution where variance depends on mean

Let $x = \bar{x} + \epsilon$ where $\bar{x} \sim \mathcal{N}(\mu,\sigma^2)$ and $\epsilon \sim \mathcal{N}(0,\sigma_\epsilon^2(\bar{x}))$ are independent, i.e., the expected value of $x$ is normally ...
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3answers
64 views

Product of random independent variables

What are the properties of the product of random variables? The book I have on probability and statistics only comments on their sum properties. I know that when you sum random independent variables ...
0
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1answer
37 views

estimates Gaussian moments

Let $X_i \sim N(0,\sigma_i^2)$. Let $k\geq0$ be a fixed integer. I would like to compute $$A:=E[|X_1-X_2|^k|X_2|^k]$$ My idea was \begin{align*} A=&\int_{\mathbb{R}^2}|x_1-x_2|^k |x_2|^k ...
0
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1answer
72 views

Does a conditional normal distribution imply an unconditional normal distribution?

I have often seen it claimed that for scalar random variables $y$ and $x$, the conditional normal distribution $$ y\mid x\sim N(0,x^2) $$ also implies the unconditional normal distribution $$ ...
1
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0answers
34 views

population of dots with normal distribution of pitch

I want to generate a plot that shows a rectangle populated with dots, where the dot-to-dot distance (pitch) distribution is a lognormal (or a gaussian). I want to be able to change the mean dot-to-dot ...
1
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0answers
8 views

Conditional Covariance of a Normal conditionally autoregressive (CAR) prior

Suppose $$\bf{\nu}|\mu,\rho,\delta^2 \sim N_p(\mu \bf{1},\delta^2(D_w-\rho W)^{-1}),$$ where $W$ is a binary symmetric matrix and $D_w$ is diagonal with $(D_w)_{ii}=\sum_j w_{ij}$, $\mu$ is a scalar. ...
3
votes
1answer
50 views

Distribution of sum of product-normal distributions.

I know that the distribution of a product $Z=XY$ of two normally distributed variates $X$ and $Y$ with zero means is the product normal distribution [Mathworld]. What is the distribution of $Q=\sum ...