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

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1
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2answers
96 views

Sum of a Normal and a Truncated Normal distribution

I have normal distribution $ N(\mu_1, \sigma_1)$ which shows the amount of demand in warehouse 1. The current amount of stock in the warehouse 1 is C. If the random demand is greater than C, it cannot ...
0
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0answers
44 views

Conditional PDF of multivariate normal distributions

Suppose that Y~$N\begin{pmatrix} 1\\ 2\end{pmatrix},\begin{pmatrix}2 & 1\\ 1 & 2 \end{pmatrix}$. How can I find the conditional PDF of $Y_1$ given that $Y_1+Y_2=3$?? I am given a hint to ...
0
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1answer
28 views

Calculating standard deviation from a set of data

I'm trying to create a normal distribution of numbers between 0 and 100. I know that the mean = 28, and the only other information about the data is that there is a 10 % change that the number is 44, ...
0
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1answer
57 views

Why is Kurtosis of ND 3?

3 seems to be an important number when it comes to kurtosis. I see that it is often removed from the value entirely and this seems to be due to its being the kurtosis of the normal distribution. I ...
1
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3answers
54 views

Finding constant for CLT normal distribution

(This is from my textbook, but I don't understand their explanation. I've Googled around, but haven't found an answer that makes sense.) $$ \mu = 0, \sigma^2 = 1, n =16 $$ Find c such that: $$ ...
1
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2answers
59 views

Determine $A$ such that $Q=X'AX$ has chi-squared distribution.

Let $\boldsymbol X\sim N_n(\boldsymbol\mu,\boldsymbol\Sigma)$, where $\boldsymbol\Sigma$ positive-definite. I am trying to determine, in general, what form $\boldsymbol A$ (one example is ...
0
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1answer
383 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
115 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
55 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 ...
0
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1answer
87 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 ...
-1
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3answers
32 views
2
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1answer
38 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 ...
0
votes
1answer
21 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 ...
1
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0answers
101 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. ...
0
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1answer
43 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
35 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
33 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
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1answer
140 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 ...
1
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2answers
52 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 = ...
1
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2answers
85 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 ...
1
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1answer
62 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
91 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
317 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
votes
1answer
155 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
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1answer
65 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 \\ ...
-1
votes
2answers
40 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\\ ...
2
votes
1answer
52 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
33 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
37 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
156 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 ...
0
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2answers
47 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
85 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
130 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{ ...
2
votes
2answers
95 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
24 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
53 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
116 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 ...
0
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3answers
94 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
votes
1answer
51 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
votes
1answer
169 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
76 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
15 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
94 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 ...
1
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1answer
266 views

Is this function increasing? (standard normal distribution, Mills Ratio)

Where $\phi\left(z\right)$ and $\Phi\left(z\right)$ represent the standard normal pdf and cdf respectively. 1) Is the function $f\left(z\right)=\dfrac{\phi\left(z\right)}{1-\Phi\left(z\right)}$ ...
7
votes
1answer
359 views

Concentration of measure bounds for multivariate Gaussian distributions (fixed)

Let $\gamma_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. It is known (see for example Cor 2.3 here: http://www.math.lsa.umich.edu/~barvinok/total710.pdf) that ...
0
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0answers
45 views

Rearrange for solving x (Miller & Siegmund, 1982; equation 8)

I have the following formula from a paper back in 1982 by Miller & Siegmund, "Maximally Selected Chi Square Statistics": α = 0.05 φ() is the standard normal density function: Everything else ...
1
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2answers
111 views

Proving completeness of the average of a random normal sample

Suppose that $n$ is a fixed positive integer and $\theta$ is a parameter belonging to $\Theta=\mathbb{R}$. Suppose that we are given that $Y_1,\ldots,Y_n$ are i.i.d. $N(\theta,1)$. I'm trying to show ...
1
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1answer
913 views

How can you normalize two data sets to the same scale?

I have two data sets, one that ranges from 0-200, and another that ranges from ~400-~2500. I would like to compare the two according to a score from 0-10. I remember about normalizing from a ...
0
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0answers
28 views

Non-Geometric Proof of Random Normal Projection Identity

Many papers on locality sensitive hashing, sketching and similar use the following lemma: If $r\in\mathbb{R}^d$ is a random vector with all entries normally, independently distributed as ...
0
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0answers
14 views

central limit theorem and sampling dist.

If you takes samples from a distribution, and you can see that they have different variances, can the central limit theorem still be applied. The computer vision teqnique i am referring to is this ...