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
34 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
votes
0answers
31 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
106 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
vote
2answers
48 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 = ...
0
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0answers
15 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
votes
0answers
56 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
vote
2answers
81 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
vote
1answer
54 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
73 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
vote
1answer
216 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
138 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
52 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
35 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
47 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
vote
0answers
36 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
133 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
votes
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
vote
0answers
80 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
vote
0answers
125 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
vote
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
vote
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
votes
0answers
80 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
votes
3answers
89 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
48 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
140 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
vote
0answers
62 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
12 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
73 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
vote
1answer
180 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
304 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
votes
0answers
39 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
vote
2answers
92 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
vote
1answer
691 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
votes
0answers
26 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
votes
0answers
13 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 ...
0
votes
2answers
71 views

Robust estimator

What does it mean that an estimator is robust? How can you tell whether an estimator is robust or not in statistics? I need to discuss whether the maximum likelihood estimators of the normal ...
1
vote
1answer
61 views

A question on a certain transformation of a normal random variable

I have to solve the following exercise: Suppose $X\sim N(\mu,1)$ and consider $Y=\dfrac{1-\Phi(X)}{\phi(X)}$, where $\phi,\Phi$ are the pdf and cdf of the standard normal distribution with ...
2
votes
1answer
42 views

Intuition for proof of Slepians Inequality

If z is a centered gaussian random variable and $ x_1 ,x_2 ,..,x_n ,y_1,y_2,..,y_n $ are points in $ \mathbb{R}^{2n} $ satisfying $ |x_i-x_j |_2 \leq |y_i -y_j |_2 \ \ \ \forall i,j \in [n] $ then $ E ...
1
vote
0answers
55 views

Show Almost Certain Convergence of a Sequence of Normal Random Variables

Let $(X_n)_{n=1}^\infty$ be independent, $N(0,1)$-distributed random variables. Prove that $$ \limsup_{n \to\infty}{X_n \over \sqrt{2 \log(n)}} = 1 \ \text{almost surely}.$$ I am aware of the ...
0
votes
1answer
115 views

Integral of normal distribution curve

I am having hoping to use the integral of the normal distribution curve to find the probability of having a mean of $0.30$ or greater, i.e. one tailed distribution. With a sample standard deviation of ...
1
vote
1answer
57 views

Standard Normal Distribution Transformation Z=lnY

I'm not sure if my approach to this problem is correct and I need help I need to apply $Z=\ln{Y}$ to the following standard normal distribution and then find the distribution of $Y$ ...
2
votes
1answer
53 views

Sum of weighted normal distributions, how to solve $P(X<x) = y$ for $x$?

How do I solve the following equation for $x$ ...
2
votes
1answer
401 views

Numerical Approximations to the Cumulative Distribution Function of the Normal Distribution

I have been trying to write the code for the Cumulative Distribution function (CDF) of the normal distribution in C++. Since the cdf does not have a closed form solution of the integral, I was ...
2
votes
2answers
42 views

Probability that $x > 0$ given that $x > y$ for independent and normally distributed $x,y$

I was recently been asked this question in an interview but not able to solve it as I am rusted in Bayesian conditional probability. Here is the question: $x$ and $y$ are independent variables ...
2
votes
1answer
43 views

Why is the $0$th percentile of the standard normal distribution $-\infty$?

Why is the $0$th percentile of the standard normal distribution $-\infty$? I can't explain the cause except saying there is no area under the curve. So it goes beyond the bell-shaped curve.
-1
votes
1answer
55 views

Independence of Gaussian Distribution Function with Different Means

Is there any way to prove that $N$ Gaussian distribution functions are linearly independent if and only if the means are different. For example, if ...
1
vote
1answer
151 views

How to derive the expected value of even powers of a standard normal random variable?

I am trying to prove that, for a standard normal random variable $Z \sim N(0,1)$, ${\mathbb E}[z^n]=n!!$ for even values of $n$. What I'm doing is integrating the p.d.f. of $Z$ which is ...
0
votes
2answers
27 views

How to analytically find probability of certain combinations between normally distributed populations?

Let's say we have two normally distributed populations A and B, which have different means and standard deviations. We then pick one item from population A and one from population B. How could we ...