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

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Confusion related to predictive distribution of gaussian processes

I have this confusion related to the predictive distribution of gaussian process I didn't get how the integration gave that result. What is P(u*|x*,u). Also how come the covariance of the posterior ...
2
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1answer
36 views

Valid distribution

If $N(0,\sigma^2)$ is the Gaussian distribution with mean $0$ and variance $\sigma^2$, is $pN(0,\sigma^2)$ a valid distribution ? $p$ is a constant and $0\le p\le 1$.
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268 views

Distribution of the $l_2$-norm of gaussian vector

Let $Y_k \sim N(\mu_k, \sigma_k^2)$. For $\sigma_k = \sigma$ the squared norm of $Y = (Y_1, \ldots, Y_n)$ follows the noncentral chi square distribution. What is the distribution in the general case? ...
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1answer
108 views

Creating an offset bell curve

This is half programming and half math, but I need the math portion answered as I'm no good at it. I have a list of 10 objects and am randomly selecting and object from that list. I need the ...
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0answers
50 views

How $\mathbb E[\bar\epsilon_{i.}-\bar\epsilon_{..}]=0$ ? $\mathbb E$ denotes expectation.

Statistical model for Complete Randomized design $y_{ij} = \mu + \tau_i + \epsilon_{ij}$ where, $i$ denotes treatment and $j$ denotes observation. $i=1,2,...,k\quad and \quad j=1,2,..., n_i$ ...
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1answer
96 views

How come $P(Z< -1.5)$ is equal to $P(Z > 1.5)$ which are both equal to $1-P(Z < 1.5)$?

I can't wrap my head around the idea they are both equal. I mean shouldn't we have $P(-Z > 1.5)$ which is not equal to $P(Z < 1.5)$?
3
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1answer
183 views

Help with gaussian integral

I need to solve this gaussian integral: $$\int_\mathbb{R} (2\pi)^{-n/2}\mid \Sigma\mid ^{-\frac{1}{2}}e^{-\frac{1}{2}(u-Kx)^T\Sigma ^{-1}(u-Kx)} u^TRu \,\mathrm du$$ It is the integral of a ...
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592 views

find probability in normal distribution

i would like to check myself if following my answer is correct: let us consider following problem: Suppose the heights of a population of $3,000$ adult penguins are approximately normally distributed ...
2
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1answer
30 views

Confusion related to matrix multiplication

I am having this simple confusion.Lets consider a multivariate gaussian distribution with mean $\mu$ and precision matrix $K$. Then the exponential term is $$(x-\mu)' K (x-\mu)$$ If I open the above ...
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2answers
187 views

Shift interval of log-normally distributed latin hypercube samples

first of all I'm not sure if this part of StackExchange is the right one because my question is mainly on a way to implement something in MATLAB. Ok, now let me try to pack my whole question in one ...
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1answer
529 views

How to merge two Gaussians

I have two multivariate Gaussians each defined by mean vectors and Covariance matrices (diagonal matrices). I want to merge them to have a single Gaussian i.e. I assume there is only one Gaussian but ...
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2answers
211 views

Probability Distribution. Case study with a bacterial population

Let's imagine, we start with one single bacterium. At each time step (generation), each bacterium has $x$ offspring and it dies (semelparous species). $x$ is a value drawn from a normal distribution ...
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1answer
106 views

Lower bound on the probability of maximum of $n$ i.i.d. chi-square random variables exceeding a value close to their number of degrees of freedom

I am wondering if there is a tight lower bound on the probability of a maximum of $n$ i.i.d. chi-square random variables, each with degree of freedom $d$ exceeding a value close to $d$. Formally, I ...
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1answer
27 views

Cumulative Normal Distribution.

Let $X_1,\ldots,X_n$ be a random sample from $f(X;\theta)=\phi_{\theta,25}$, that is, $X_1,\ldots,X_n$ be normally distributed with mean $\theta$ and variance $25$. I am not understanding how ...
2
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1answer
624 views

Expected Value of Normal CDF

I am trying to calculate the expected value of a Normal CDF, but I have gotten stuck. I want to find the expected value of $\Phi( \frac{a-bX}{c} )$ where $X$ is distributed as $\mathcal{N}(0,1)$ and ...
3
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0answers
88 views

Exponentials of chi-squared random variables (and their sums)

Let $X_1,X_2,\ldots,X_n$ be a sequence of i.i.d. chi-squared random variables with $t$ degrees of freedom, i.e. $X_i\sim\chi^2_t$. I am wondering what is known about the distribution of ...
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2answers
316 views

Distribution of $Y = \sin X$ when $X$ is normal?

Assume $X$ is Normally distributed : $X\sim N(\mu,\sigma)$ What is the distribution of $Y = \sin X$ ? I think we should start with $F_Y(y)=P(\sin X < y)$. But how to continue?
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0answers
76 views

Is there an algebraically normal function from $\mathbb{Z}^{2}$ to $\{ 0 , 1\}$?

Let $\gamma : \mathbb{R} \to \mathbb{R}^{2}$ be a real algebraic curve. Let $r \geq 0$ and $I \subset \mathbb{R} $ then $\gamma_{r} (I)= \{a \in \mathbb{R}^{2} : \exists b \in \gamma(I), d(a,b)\leq r ...
2
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1answer
112 views

Weighted integral of random variables

Given a random zero-mean gaussian random variable $X(t)$ with parameter $t$, such that $E [X(t) X(t^\prime)] = \sigma^2 (t) \delta_{tt^\prime}$, is it possible to produce a single gaussian random ...
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3answers
187 views

How to integrate the difference between the CDFs of two normal distributions

I have two normal distributions A and B. I am trying to write a program that will take mean(A), stddev(A), mean(B), stddev(B) and output the result of the following equation: $$ ...
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1answer
49 views

Determine which mean is smaller over two non-normal distributions

Let's say I have a non-normal distribution A and another non-normal distribution B, the mean and std deviations of each distribution are different. I then randomly sample 100 values from A, SampleA, ...
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3answers
458 views

Is the mean of the truncated normal distribution monotone in $\mu$?

I am wondering whether the mean of the truncated normal distribution is always increasing in $\mu$. The untruncated distribution of $x$ is $\mathcal{N}(\mu,\sigma^2)$. The mean of the truncated ...
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0answers
614 views

Uniform distribution on the surface of unit sphere

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere. Intuitively, I know that that's ...
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0answers
63 views

iterative transform of standard normal random variable

Given a discrete series of random variable $n(i)$ that each element follows the standard normal distribution $N(0,1)$, another series is defined iteratively as: $$u(i+1)=au(i)+bn(i)$$ where ...
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0answers
754 views

calculate probability without table

my question is related to normal distribution,namely as i know in GRE quantity section,there could be question related to normal distribution,but of course we will not have table,o how can we ...
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2answers
62 views

Noise pdf Gaussian

Why the probability distribution function of the noise in a channel is Gaussian (normal distribution)? Intuitive discussion is appreciated.
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1answer
55 views

Finding confidence interval for a binomial process using the normal distribution?

See, when I was taught how to find confidence intervals, I always needed the sample variance to use a Student $t$ distribution to form the confidence interval. How does this work in the binomial case ...
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2answers
985 views

Mean and variance of the product of a normally distributed random variable

If a random variable X is normally distributed: $X \sim N(\mu,\sigma^2)$ what is the mean and variance of the random variable $Y = aX + bX^2$, where $a$ and $b$ are constant Given that ${\mathbb E} ...
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1answer
297 views

Going from the Poisson distribution to the Gaussian.

In this lecture, at about the $37$ minute mark, the professor explains how the binomial distribution, under certain circumstances, transforms into the Poisson distribution, then how as the mean value ...
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0answers
49 views

Calculating Expectation

I want to verify the following equation: $$E[(xe^{aY-\frac{1}{2}a^2}-b)^+]=x\Phi(l_1)-b\Phi(l_2)$$ where $Y\sim \mathcal{N}(0,1)$, $\Phi$ the distribution function of a standard normal distribution, ...
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1answer
381 views

Why do we use a $z$-test rather than a $t$-test when estimating an appropriate sample size?

I'm kinda puzzled on one point. In our stat class, we are taught to use the Student $t$ distribution to find confidence intervals for normally distributed data, as blindly using the normal ...
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1answer
236 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
3
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1answer
606 views

Convergence of binomial to normal

Problem: Let $X_n \sim \operatorname{Bin}(n,p_n) $ where $p_n \xrightarrow{} 0$ and $np_n \xrightarrow{} \infty$. What I need to show is that $$\frac{X_n - np_n}{\sqrt{np_n}} \xrightarrow{d} N(0,1) ...
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0answers
98 views

Normal distribution inequality

Let $n(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) = \int_{-\infty}^x n(t)dt$. Prove the following inequality. $$(x^2+1)N + xn-(xN+n)^2>N^2$$ where the dependency of $n$ and $N$ on ...
2
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1answer
226 views

3-D generalization of the Gaussian point spread function

I would like to extend to 3-D the formulation of the 2-D Gaussian PSF, given by: $$k_{\sigma}(x,y)=\frac{1}{\sqrt{(2\pi)^2}\sigma^2}\exp\left[-\frac{x^2+y^2}{2\sigma^2}\right]$$ Is the following 3-D ...
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2answers
110 views

Lower bound on the probability that the maximum of a sequence of $n$ i.i.d. standard normal r.v.'s exceeds $x$

Let $X_{\max}=\max(X_1,X_2,\ldots,X_n)$ where $n$ is large and each $X_i$ is i.i.d. standard normal random variable, i.e. $X_i\sim\mathcal{N}(0,1)$. Is there a lower bound on the probability ...
3
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2answers
53 views

Can $\Phi^{-1}(x)$ be written in terms of $\operatorname{erf}^{-1}(x)$?

Can the inverse CDF of a standard normal variable $\Phi^{-1}(x)$ be written in terms of the inverse error function $\operatorname{erf}^{-1}(x)$, and, if so, how? This seems like an easy question, but ...
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3answers
412 views

Why is normal distribution more accurate than binomial distribution?

I'm having a tough time understanding this. This is what I am told about comparing the two: The probability that Saredo is late for school is 0.6. What is the probability that in one month she is ...
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1answer
42 views

Normal Distribution and a Discrete Amount

From what I understand about normal distribution is that you make a discrete number continuous by adding .5 which every way the question asks for. What if you were to have a discrete number with a ...
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1answer
215 views

Convergence in distribution and standard normal distribution

Let $X_1,X_2,\ldots$ be independent random variables with $X_k$ distributed as $\mathcal{N}(0,1)$ and $S_n=X_1 X_2+X_2 X_3+\ldots+X_n X_{n+1}.$ Show that $\frac{S_n}{\sqrt{n}}$ converges in ...
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74 views

Proof of theorem, multivariate normal distribution

The following theorem was presented in my textbook without proof and I would be thankful if someone could refer me to a proof of it: Suppose that $ \boldsymbol{X} \sim ...
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1answer
383 views

What is the difference between distribution and dispersion?

I need to explain the difference between a distribution (Normal, Chi-square, Poisson, etc.) and Dispersion (as measured by variance, standard deviation) to some students. What is the simplest ...
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1answer
71 views

How to normalize histogram well?

UPDATE 2 The question may be formulated as follows: Is there any common probability distribution, like normal distribution, but which has sharp (or just sharper) edges? If yes, then I could ...
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1answer
108 views

Normal random Vector

Question: Prove that linear functions of the form $\bar{y}=\bar{b}+\mathrm{B}\bar{x}$ are normal random vectors provided that $\bar{x}$ is a normal random vector. Find $E(\bar{y})$ and $V(\bar{y})$. ...
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1answer
149 views

Find the standard deviation of $ \frac{\gamma}{\sqrt{2\pi\sigma}}\exp\left(-\frac{\gamma^2}{\sigma}\frac{(x-\mu)^2}{2}\right)$

Given $\frac{\gamma}{\sqrt{2\pi\sigma}}\exp\left(-\frac{\gamma^2}{\sigma}\frac{(x-\mu)^2}{2}\right)$ as a normal distribution PDF with mean $\mu$, I'd like to solve for the std deviation in terms ...
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1answer
156 views

Normal Distribution Probability

At a large publishing company, the mean age of proofreaders is 36.2 years, and the standard deviation is 3.7 years. Assume the variables are normally distributed. a. If a proofreader in the company ...
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1answer
171 views

Characteristic function of vector-valued random variables

I just begins my self-study on Brownian motion. I got stuck on the part about random-vector and characteristic function. Here are my questions: I'm not quite get about how characteristic function of ...
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2answers
224 views

Computing the Gaussian integral with step functions

Say, we are interested in deriving $$\int_{-\infty}^{\infty}e^{-x^2}=\sqrt{\pi}\tag{1}$$ There are many well known ways to do it, for example: by polar coordinates via the gamma function, etc. ...
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1answer
96 views

To what extent the statement “Data is normally distributed when mode, mean and median scores are all equal” is correct?

I read that normally distributed data have equal mode, mean and median. However in the following data set, Median and Mean are equal but there is no Mode and the data is "Normally Distributed": $ 1, ...
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3answers
270 views

Central Limit Theorem Definition

My friend and I have a bet going about the definition of the Central Limit Theorem. If we define an example as a number drawn at random from some probability density function where the function has a ...