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

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8 views

Finding normal curve given the minimum and maximum - is it possible?

I have a quick question regarding the normal distribution, or really any kind of distribution as it can also be skewed if need be. I was wondering if it were possible to let the curve have a minimum ...
0
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0answers
26 views

A Gaussian Divided by a Gaussian Equal to A Gaussian Divided by a Constant

I have a neural-network model in which each neuron is associated with an angle $\theta$. Firing rate as a function of $\theta$ is either a Gaussian or a constant. The claim has been made using this ...
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1answer
17 views

Pdf of a normal variable accepted with probability dependant on the normal variable

Assume $z$ is a standard normal variable. If $z<0$, then we accept it with probability 1. if $z\ge0$, we accept it with probability $e^{-mz}$, where $m>0$. I'm trying to figure out the pdf of ...
6
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1answer
56 views

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - ...
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0answers
11 views

Variance of a quadratic form

I am considering a variance of two forms: $ R(x) = (x-m)^\top A (x-m) + b^\top (x-m) + c $ $ R'(\Delta) = \Delta^\top A \Delta + b^\top \Delta + c $ where $x$ is a random variable of ...
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1answer
8 views

Proporties of linear-combinations of normally distributed variables.

At my school, to pass an exam, you'll have to score at least 230 points. The results are normally distributed with $\mu=200$ and $\sigma=20$. If I were to consider 10 students who attends the exam, ...
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0answers
17 views

Height of a point

I wonder if there is way to find out the height of a specific point x in a normal distribution whenever the standard deviation is not given.?
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0answers
26 views

Find the distribution of $Z = 1/X_1 + 1/X_2$

Find the distribution of $Z = 1/X_1 + 1/X_2$, where $X_1$ and $X_2$ follow normal distribution. I have $2$ variables with normal distribution, $X_1$ and $X_2$. How can I find the distribution of: ...
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2answers
32 views

Proving the $Pr(d>0|a+d=\pi)$ is increasing in $\pi$ when a and d are two independent normal distributions.

I was wondering if it is possible to prove the following (or show false otherwise). Given two independently distributed random variables $a\sim \mathcal{N}(\alpha,\sigma_\alpha^2)$ $d\sim ...
1
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1answer
18 views

Translate exponential distribution into normal distribution

I have a bunch of inventory management formulas that are supposed to be used with normal distributions, however my demand data fits an exponential distribution. Is there any way to translate the ...
0
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0answers
9 views

Exponential Demand Periodic Review

I have exponentially distributed demand data and I am trying to find a formula for an 'order up to level (OUL)' periodic review ordering policy. We are not using a re order point for this policy. ...
1
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0answers
16 views

Suggestions for dealing with these order statistics

Consider a collection of $n$ random variables $X_i \sim N(\mu, \sigma^2)$, ($i = 1,2,\ldots, n$) and a random variable $X \sim \text{Exp}(\lambda)$. All $X_i$'s and $X$ are mutually independent. Let ...
4
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1answer
62 views

Why has the Stein operator for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$?

My Question: Why has the Stein operator $\mathcal A$ for normal approximations the form $(\mathcal Af)(x)=f^\prime(x)-xf(x)$? How can one deduce this form of the operator? Reason for my question: I ...
1
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0answers
15 views

Calculation of arrival time of messages from 1 source through 2 different routes

I need to simulate sending messages from $A$ to $B$ as follows: Each message is sent $N$ times from $A$ on the same time, passes through a certain route $R_n$ and arrives at $B$. Travel time of $R_n$ ...
2
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0answers
25 views

normal squared characteristic function derivation

I'm trying to derive the normal squared characteristic function, there's already a question on this but the answer has a part which is "proved as an excercise" which I try to do here. Is my proof ...
1
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1answer
33 views

Minimum matching convolution

Let $\text{SPD}^n$ and $\text{PD}^n$ be the semi-positive and positive definite matrices in $\mathbb{R}^{n\times n}$, respectively. I want to find an $X\in \textrm{SPD}^n$ that minimizes $||X||$ ...
0
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1answer
13 views

Variable drawn from a normal distribution

What exactly is the meaning of a "variable drawn from a normal distribution"? I know what a normal distribution is, but my main exposure to "variables" is from calculus, so I have a hard time ...
2
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1answer
510 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 ...
2
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3answers
41 views

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian.

If two different linear combinations of two random variables are Gaussian, can we deduct both of them are Gaussian. Mathematically, if we know that $a_1X+b_1Y$ and $a_2X+b_2Y$ have Gaussian ...
0
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0answers
20 views

Mixture of binomial distributions

I have a population of agents with a single real-valued attribute $x$. Each of them performs $n$ Bernoulli trials with success probability $q(x)$ which depends on their attribute. In particular, $$ ...
0
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1answer
12 views

Reconstructing a restricted distribution from its mean and standard deviation

For simplicity lets assume we have a continuous distribution from 0 to 100. If the mean is 60 and std is 10, then it would make sense to simply model it as a gaussian with those parameters. However ...
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1answer
335 views

Moment Generating Function of Gaussian Distribution

Derive from first principles, the moment generating function of a Gaussian Distribution with $$PDF= \dfrac{1}{\sqrt{2\pi \sigma^2}}e^{-(x- \mu)^2/2\sigma^2}$$ MY ATTEMPT MGF= E[$e^{tx}$]= ...
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0answers
15 views

I would like to know how to do log transformation of hyperparameters in Gaussian Process Classification.

I am using Gaussian Process classification and I want to do log transform of the hyperparameters so that they are all positive. From this www.lce.hut.fi/research/mm/gpstuff/GPstuffDoc.pdf document, I ...
2
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1answer
529 views

Expected Value Problem (Q-function…inside a function)

I'm working through my textbook for a communications course I'm taking, and this problem is confusing me big time. Like always, the math questions give me the most problems. Maybe I should take the ...
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0answers
15 views

Confidence interval from covariance matrix

We have a matrix of stochastic variables $X\sim\mathcal{N}(0,\Sigma^2)$, where $\Sigma^2$ is a positive definite covariance matrix. How do we calculate the 95% confidence interval for X? (lets say ...
2
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1answer
40 views

Are $X$ and $Y$ necessarily normal if the the sum $Z=X+Y$ is normal?

Of course, asking the question the other way round is straightforward to answer as via the convolution we find that the sum of two normal distributed variables is again normal. But however, is it ...
1
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2answers
34 views

Normal Distribution of Sums

I have two normally distributed random variables $X$ and $Y$. Then I know that the sum $X-Y$ is also normally distributed (i). However, I want to show (preferably by a counter example) that the ...
0
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0answers
16 views

Statistics, distributions and graphing

Lets say that you have data containing one variable - the waiting time between each car passing from 1200 hours... so you have something like 23, 54, 26, 8, 2, 59 etc what is the best way to analyse ...
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0answers
30 views

Monotonocity of ratios of normal CDFs

I am solving a problem in decision theory under uncertainty and need to establish whether $\frac{\Phi(x)-\Phi(x-\varepsilon)}{\Phi(x+\varepsilon)-\Phi(x-\varepsilon)}$ $(\ast)$ is monotonically ...
1
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1answer
71 views

conditional expectation of squared standard normal

Let $A,B$ independent standard normals. What is $E(A^2|A+B)$? Is the following ok? $A,B$ iid and hence $(A^2,A+B),(B^2,A+B)$ iid. Therefore we have $\int_M A^2 dP = \int_M B^2 dP$ for every ...
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1answer
38 views

chi square distribution probability

I am having a problem with this. Suppose a stock's returns are normally distributed with mean $m$ and variance $\alpha^2$ and we compute the sample variance from a sample of $41$ periods and find ...
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2answers
19 views

What is a decision threshold and how does it apply to a statistical power?

I'm pretty confused on what is actually going on in this section with hypothesis testing. As another note, the values below are computed using R. I have a homework problem that says: From the ...
1
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1answer
422 views

Simulation of a Gaussian process on $R^2$ with a stationary kernel using the Karhunen-Loève expansion

Assume $X(\omega, t) \sim \mathcal{N}(0, K(\cdot, \cdot))$ is a real-valued, centered Gaussian process on $R^2$, i.e., $X: \Omega \times R^2 \to R$. Let the covariance function of the process be ...
0
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1answer
29 views

Is the product of $n$ Normal distributions also a normal distribution? [closed]

Is the product of $n$ Normal distributions also a normal distribution? I need not derivation just but answer.
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0answers
7 views

convergence multivariate normal

If $X$ and $Y$ have asymptotic normal distribution then using Slutsky's theorem $aX+bY$ is also asymptotic normal, can I conclude that the vector $(X,Y)$ is asymptotic bivariate normal? If not, how ...
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0answers
31 views

What is your favorite approximation to the normal distribution? [closed]

I am asking this because my favorite is this one, which I independently discovered: $$N(x, 0, 1) =\frac1{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2} dt \approx \frac{1}{1+\exp(-ax)}, \text{ where } ...
0
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0answers
19 views

R Studio help - qqnorm [closed]

I'm loading in a file called data and using the command qqnorm like this data <- read.table("data.txt", header=TRUE) qqnorm(data) And get the error message "Error in xy.coords(x, y, xlabel, ...
3
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2answers
30 views

Normal approximation of binomial distribution - limits

In binomial distribution number of successes (usually denoted as $x$) must be between between $0$ and $n$, inclusive ($n$ is the number of trials). So for example there can be a problem which asks for ...
1
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0answers
19 views

Distribution of unknown covariance matrix, given variance of linear combination

Suppose I am uncertain about the covariance of a vector-valued random variable $X$, but the variance of some linear combination is known. How does this affect the distribution of $X$? Specifically ...
1
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1answer
12 views

When the domain of a continuous distribution exceeds feasible values, what should I do?

Now I need a (maybe approximated) model for this distribution: $$X=(x_1, x_2, …, x_n)$$ where $x_i$ is a real number between $0.0$ and $1.0$, and the sum of $x_i$ equals $1.0$. Now, I want to use ...
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0answers
23 views

The mean deviation from mean in a normal distribution is equal to $4\sigma/5$ [closed]

Show that the mean deviation from mean in a normal distribution is equal to $4\sigma/5$. Progress. I have tried going by the usual definitions of deviation and mean deviation but am stuck. Tried ...
1
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0answers
17 views

MLE of variance for a spherical Gaussian

I am trying to implement the X-Means clustering algorithm. In it, the authors use the BIC to determine which model fits the data best. It is explained here: ...
0
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1answer
275 views

Calculating P(A>B), where A and B are normal distribution

In the problem we have that $A \sim N(7, 11/60)$ and $B \sim N(7.3, 7/20)$ and the question is what is the probability that $A$ gives a higher value that $B.$ Since the textbook we have for the course ...
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2answers
86 views

Why does normal distribution have the same shape regardless of its parameters?

The formula for normal distribution is quite complicated, it has $\sigma$ in the exponent and in denominator. And no matter what $\sigma$ is, the shape of its pdf is the same (i.e. for example 3 ...
1
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1answer
350 views

Expected value vs using method of indicator

I am having a hard time understanding the difference between getting the Expected value by finding the mean E(X)=np and using the method of indicator to find the expected value. For example if we ...
0
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1answer
292 views

Gaussian distribution and its parameters

I need to learn more about Gaussian distribution and given a set of data, plot a Gaussian distribution of it. Using the following code sample, could you please tell me how I can plot a Gaussian ...
1
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0answers
24 views

How good of an approximation is a normal probability distribution for sum of dice rolls?

I want to know how well the normal distribution explains the sum of rolls with n dice with s sides. The mean value and the variance of the dice rolls are $$\mu=n\frac{s+1}{2}$$ and ...
9
votes
1answer
119 views

Characterization of normal distribution

I am sorry if this question is vague since I am completely unfamiliar with probability theory. Suppose that we have a family of real-valued random variables $X_n$ (say, all of them have mean 0) on ...
3
votes
1answer
439 views

How to plot standard deviation rings of a bivariate normal distribution?

I'm working on a project right now where I have Gaussian distributions, and I want to create a graphic that represents them. I'm not sure how to generate the ellipse that represents say 1 standard ...
1
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2answers
17 views

Formula to get mean and standard deviation of this multi-variable equation

$$ \binom n x \times\left(\frac1r\right)^x\times\left(\frac{r-1}r\right)^{n-x} $$ If you have $n$ boxes and have a $\frac1r$ chance to fill each one, this equation returns the chance that you fill ...