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

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1answer
44 views

“Trick” to demonstrate expression is a probability density function for the Gaussian Distribution. [closed]

I was looking into a particular method to demonstrate that the following expression is a probability density function for the Gaussian / Normal distribution. (i.e. that the integral = 1) : $$ f_{X} ...
1
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2answers
25 views

Is Z in standardized normal distribution always positive?

The question asks me to find the mean, given that: $\sigma$ is $0.8$ when not standardized 96% is over 40 I worked out that $Z = -1.75$ in this case, which then would lead me to $$Z = ...
0
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1answer
24 views

Non-integrability of the pdf of a squared normal random variable

If $X$ is a normal random variable with mean $0$ and variance $1$, then the pdf of $Y=X^2$ is $f_Y(y)=(2\pi y)^{-\frac{1}{2}}e^{-\frac{y}{2}}$ (for $y\ge0$). But $f_Y(y)$ is not integrable at $0$, ...
0
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0answers
14 views

Quantile computation for a Bivariate Normal Distrubtion

Is there a way to compute the following Quantile $x$ since $a ,b$ are known: $$P(T_1 < a , T_2 > x ) = b $$ $T_1$ and $T_2$ have a bivariate normal distribution with known mean and covariance. ...
2
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0answers
42 views

Ammunition Depot: Monte Carlo Method

I was given the following question from a friend of mine and I can't seem to understand it to well: A squadron of 10 bombers attempts to destroy an ammunition depot. The fighter jet flies in the ...
3
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1answer
59 views

AWG Noise and RMS Voltage

A question says, a channel is corrupted by Additive White Gaussian Noise with zero mean and RMS voltage 20 nV. The probability that the noise voltage is less than a particular positive value c is 0.9. ...
0
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0answers
12 views

Manipulation of this equation into a Gaussian form

I have been going through a paper (http://www.jting.net/pubs/2007/ting-ICRA2007.pdf) and trying to work out the maths. Ultimately, I came to the following expression $$ \bigg[\frac{1}{2 ...
1
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1answer
35 views

Normal distribution random variable engine blocks

The diameters of cylinders drilled into an engine block vary slightly, being normally distributed with a mean of 12.500 cm and a standard deviation of 0.002 cm. If the diameter of a given cylinder is ...
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0answers
9 views

Finding the distribution of the Sample mean of normal distribution (check my working?)

If $X_1,X_2,...,X_n$ are iid $N(\mu, \sigma^2)$, then how do we find the distribution of the sample mean $\bar X = (1/n) \sum_{i=1} ^n X_i$ ? I tried this: $ \bar X \sim N(\sum_{i=1}^n \mu_i/n, ...
1
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1answer
38 views

Normal distribution random variable probability

The daily exchange rates for the five-year period 2003 to 2008 between the euro (EUR) and the British pound (GBP) are well-modeled by a normal distribution with mean 1.459 euros (to pounds), and ...
0
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1answer
20 views

Modeling Gaussian components with standard vs exact functions

I'm studying a paper on modeling DNA histograms. It presents two alternative formulas for modeling Gaussian components: Standard form: $G(x) = ...
0
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0answers
37 views

L2 norm of 2 Normally distributed variables

Given: $Z=\sqrt{X^2+Y^2}, X\sim N(\mu_x,\sigma_x^2), Y\sim N(\mu_y,\sigma_y^2)$ What is the expected value of $Z$? I'm specifically looking for the case where the $\mu_i$ are non-zero and $\sigma_i$ ...
0
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1answer
33 views

Difference between gaussian and log-normal distribution

I have a random variable say X that is a Gaussian distributed with mean equal to zero dB. When I convert it into linear domain, i.e from dB to linear, does it imply that the resulting variable is ...
0
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1answer
28 views

Normal Distribution Statistics

I am trying to understand normal distribution, and I am trying to get Verbal SAT scores following the normal (430, 100) distribution. What is the middle range of scores encompassing 50% of the ...
1
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2answers
64 views

conditional probability of a sum or iid normal random variables given a bound on a subset of them

Let $X_i$ be iid normal random variables with mean 0 and standard deviation $\sigma$. Is there a straightforward formula to compute the conditional probability $\mathbb{P}(\sum_{i=1}^{k}X_i < ...
0
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0answers
16 views

Probability Distribution of symmetrically weighted sum of Bernoulli Random Variables $ R = \sum_{j=1}^{N} a_{j}b $

Consider $ R = \sum_{j=1}^{N} a_{j}b $ where b is a bernoulli random variable with $Probability(b=1) = p$ and $a_{j} \in \mathbb{R}^2$ is a vector . There is a line $Y = mX$ such that $ ...
0
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2answers
56 views

PDF of several draws from an uniform distribution?

Suppose I draw several times from an uniform distribution, $X\sim\mathcal{U}(0, 1]$. (I'll use $\mathrm{R}()$ to denote an independent drawing.) What is then the PDF of several draws, added and/or ...
1
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1answer
59 views

Difference between the Error function and Normal distribution function?

I have just started reading about the Error function but Distribution theory is not my strong point. So I apologize in advance for asking really simple questions about it. So the Error function is ...
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0answers
20 views

Conditioning multivariate Gaussian on a function of coordinates

I have a pretty general question and I would really appreciate if you give me any hints or point me towards some relevant literature. Suppose $X$ is an $n$-dimensional Gaussian vector. What is the ...
0
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1answer
20 views

normal distribution with mean and standard deviation

In an examination the number of marks allotted to each candidate is an integer. If the marks were normally distributed, and the distribution had a mean of 45 and a standard deviation of 12, find the ...
6
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1answer
75 views

Is the sum of a Wishart matrix and a deterministic psd matrix “almost Wishart”?

Let $XX^T$ be a Wishart matrix, generated by taking the columns of $X$ to be i.i.d. standard $p$-variate normal vectors. Let $AA^T$ be a non-random positive definite matrix. Though it is not possible ...
0
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1answer
22 views

Multivariate Gaussian vs univariate

I am developing a classification task where I generate a gaussian distribution (its mean and standard deviation) from a set of 3-dimensional data. My question is: would the classifier give same ...
0
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0answers
32 views

Determining the uncertainty from a reduced chi-square test

The questions wants me to generate 4 random numbers from a standard normal distribution (mean=0 and $\sigma$=1 ) and then calculate the sum $$x_1^2+x_2^2+x_3^2+x_4^2$$ If I do this 1000 times, how ...
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1answer
84 views

Estimate the mode of the binomial distribution without Stirling's formula

Context: Let $\tilde B_n$ be standardized binomial distributed with $p\in(0,1)$ be the probability of success in the $n$ Binomial trials. So $P(\tilde B_n = x_k)=\binom nk p^k q^{n-k}$ for $x_k = ...
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0answers
22 views

Possible prior distributions of the variance of a normal

I consider a normal distribution with random variance. Typically, in this setting the variance is modelled using the gamma prior distribution. However, for reasons of further analytical tractability ...
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0answers
34 views

Joint and Conditional pdf of Two Normal r.v.'s

Given two uncorrelated Gaussian random variables, X ~ $\mathcal{N}(0,1)$ Y ~ $N(0,1)$ Find $f_{y|x}(y|x)$, $f_{x,y}(x,y)$ If I can find either the conditional probability or the joint ...
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1answer
25 views

Determining the distribution of Y based on $x̄$ and $S^2$

Let $X1,...,Xn$ be a random sample from $N(μ,σ^2)$ and let $x̄$ and $S^2$ be the usual sample mean and sample variance. Define the random variable $$Y=c(x̄-μ)^2/S^2$$ Find c such that Y is a "named" ...
0
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1answer
35 views

Probability for membership in different normal distributions

i have following problem which I am working on: Say I want to measure something to find out whether it is more likely to belong to class A or class B. for class A: p = 1 for class B: p = 5 I ...
0
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1answer
39 views

Find the density of the random variable $Y=X^2$

Find the density of the random variable $Y = X^2$, if the random variable $X$ follows a standard normal distribution. I think I should use mgf to solve it is that right ? what should I do to start ?
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0answers
9 views

Principal Component Analysis in a stable framework

are you familiar with stable distributions. It is denoted by $S_{\alpha}(\sigma,\beta,\mu)$ where $\alpha$ is the tail index, $\beta$ is the skewness, and $\sigma$ and $\mu$ are the location and scale ...
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0answers
13 views

Finding intersections of two normal distributions

Given two normal distributions that describe each the posterior probabilities of a class A, i.e P(A|x) and B, i.e P(B|x). How do I express the decision boundaries (intersection points) as function(s) ...
1
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1answer
13 views

Normal distribution problem with expectation and deviation values not known.

I am trying to solve the following problem. The diameter of a product manufactured can be regarded as a random variable E. During an inspection products with a thickness greater than 1.01 mm ...
0
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0answers
15 views

Z Score Alternative for Asymmetric Distribtions

I would like to find the Z score for an asymmetric distribution. Problem with using the standard formula is that it is for normal distributions. If used in my case, the generated score for a given ...
4
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1answer
38 views

Application of Stein's Lemma to Calculate Moments of Normal (0,1)

Say we have $X \sim N(0,1)$. I was wondering how we can use Stein's Lemma to derive the moments of the r.v. $X$ by calculating the first few moments. How would you summarize it in the form $EX^n$ ...
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0answers
25 views

Does the standard deviation of a normal distribution change when looking at a shorter period?

Say we are given a normal distribution with mean $\mu$ and standard deviation $\sigma$, where $\mu$ is the expectation in time period $T$. Now, say you need to use the distribution over a shorter ...
0
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1answer
88 views

Computing the proportion of a normally distributed population between specified values of the standard deviation

Ok I am stuck on the second part of this question. (a) what percentage of the population is within ± $0.5$ standard deviations of the mean. (b) what percentage of the population is more than 1 ...
3
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1answer
43 views

Hölder type of inequality?

Is the following inequality true? I can't find a counterexample so I start to believe it is true but I do not manage to prove it :) Any ideas? Let $f$ be a compactly supported bounded twice ...
1
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1answer
17 views

Meaning of numbers in covariance matrix

For a course I'm following we need to work with multivariate guassians. In this case there are four variables $x_1$ through $x_4$ with the specified covariance matrix. Even though the matrix is ...
1
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1answer
47 views

Confidence Interval problem

Which of the following will result in a wider confidence interval? Check all that apply. ...
0
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0answers
17 views

Distribution of ratio of 2 points drawn from normal distribution?

Let's say we have a known normal distribution $N(\mu,\sigma^2)$. I now draw 2 points $p1$ and $p2$ randomly from this Gaussian distribution for every observation, and repeat this process large number ...
0
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0answers
163 views

AP Statistics Chapter 6 MC Review (TOPIC: Combining Normal Random Variables)

“Insert tab A into slot B” is something you might read in the assembly instructions for pre-fabricated bookshelves. Suppose that tab A varies in size according to a Normal distribution with a ...
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1answer
46 views

The ratio of WHAT type of independent random variables is normal?

I know that the ratio of two independent normal random variables is a Cauchy random variable. The ratio of WHAT type of independent random variables is normal?
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0answers
18 views

Can't figure out problem using joint density with bivariate normal distribution.

Not even sure where to start with part (b) for the problem below. For part (a) assuming the worker knows her own skill level and the prevailing wage, I got: y1 > y 0, or S1 - S0 > ln(w0/w1) for ...
0
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1answer
30 views

Characteristic functions of Poisson and normal distribution

Basically the question is in two parts: $1.)$Finding the characteristic of $P{(\lambda)}$, and it is given, I just do not know how to get the sum that they got in the very last step in this ...
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0answers
9 views

How to calculate the complexity (via Fisher information) for Fechner's and Steven's models?

In the paper on Counting probability distributions [...], the authors derive the functional complexity of two models. (My apologies, the paper requires journal access due to copyright laws.) I think I ...
1
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1answer
16 views

Efficient methods for drawing random numbers and Monte Carlo for Tsallis q-Gaussians

I would like to draw random numbers from the q-Gaussian used in "Tsallis statistics." This is specifically the distribution $$ f(x) = {\sqrt{\beta} \over C_q} e_q(-\beta x^2) $$ where $$ e_q(x) = ...
0
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1answer
24 views

If $Y\sim N(\mu T,\sigma^2 T)$ then $Y=\mu T+\sigma \sqrt{T} Z$ where $Z\sim N(0,1)$

Let $S_T$ be the price of a traded asset at time $T$. Also let: $\ln(\frac{S_T}{S_0}) \sim N(\mu T,\sigma^2 T)$ My question is, how is it that: $\ln(\frac{S_T}{S_0})=\mu T+\sigma \sqrt{T} Z$ where ...
2
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1answer
50 views

The expected value of an order statistic for normal random variables

Let $X_1$ and $X_2$ be a random sample from normal distribution with mean equal to zero and variance $\sigma^2$. Prove $E[X_{(1)}]= \frac{-\sigma}{\sqrt{\pi}}$. May I have to standarize the sample? ...
0
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1answer
19 views

Conditional gaussians, particular calculation

I'm looking for confirmation that my solution to this problem is correct. The result seems unintuitive. Given $\{X_i\}_{i = 1}^{10}$ $0$ mean jointly gaussian RVs with $\mathbb{E}X_iX_j = 2^{-|i - ...
2
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1answer
30 views

$2^\text{nd}$ Derivative of normal distribution, evaluated at one standard deviation

What is the $2^{nd}$ derivative of the normal distribution at one standard deviation? The normal distribution is given by $N(x,\mu ,\sigma)=\frac{1}{\sigma\sqrt{2\pi ...