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

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Generate Correlated Normals

I want to generate normals $X,Y,Z$ with the correlation matrix $R$ but with means $0, 1, 2$ and variances $4, 16, 25$ respectively. How can I do this? Is it possible to apply Cholesky?
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1answer
46 views
+50

Simulate two centered normal random variables with given variances and given covariance

How can I, by the central limit theorem, simulate two random variables $Z_{1}$ and $Z_{2}$ such that $$Z_{1}\sim N(0,\sigma^{2})\qquad Z_{2}\sim ...
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0answers
16 views

Differentiating unimodal and bimodal normal distributions

I have a large number of data sets that have either a unimodal normal distribution or a bimodal normal distribution. I'm not a statistician by any means, so I'm quite limited in my experience. For ...
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0answers
5 views

Variance of truncated multivariate Gaussian

Let $X \in R^n$ be distributed as the standard multivariate Gaussian i.e. $\mathcal{N}(0,I)$. I want to understand the covariance of the distribution conditioned on certain sets. Let $P_S$ be the ...
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1answer
26 views

How can I express this in terms of Gauss-Hermite Quadrature?

I am having the following expression. This is the PDF of Nakagami-Lognormal Distribution. I want to express in terms of Gauss-Hermite abscissas and weights. How can I do it? ...
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0answers
12 views

How to determine parameters of a normal distribution from a limited range of points?

In an experiment my data points are almost normally distributed with meanvalue != 0. My problem is I can only detect positive points (located on the right side of y ...
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2answers
23 views

How to compute approximate probability with z value?

This is the question: An airplane with room for 100 passengers has a total baggage limit of 6000 lb. Suppose that the total weight of the baggage checked by an individual passenger is a random ...
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1answer
866 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 ...
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0answers
33 views

Integrals with erf^N

Can anyone help with integral of type. In general, what to do if erf is in power higher than 1? $$g(S|S<L)=\frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{+\infty} \left [ \frac{1}{\sqrt{2 \pi ...
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0answers
17 views

Non-uniform convolution with discrete wavelet transform

I understand that if you have a circular N-dimensional convolution matrix, it can be diagonalized with the Fourier transform of the convolution operator. This makes it easy to calculate the density of ...
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0answers
31 views

Product of matrix-valued normal densities and Kronecker product

I am trying to find an expression for the mean, column-covariance and row-covariance matrices of the product of two matrix-valued Normal distributions. Here is what I've tried in a special case I ...
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12 views
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35 views

If this mean time is estimated to be in excess of 7 days, a new process will be implemented to reduce production costs.

a) You have just graduated with a post graduate degree in business and have obtained a position with a large manufacturing firm. The director of marketing has asked you to estimate the mean time ...
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67 views

i. If a sample of 25 contracts is selected, how likely is it the sample will overestimate the population mean by more than $10 million? [on hold]

Government officials in Canberra have recently expressed concern regarding overruns on military contracts. These unplanned expenditures have been costing Australians millions of dollars every year. ...
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17 views

The prime minister will accept an error of $5 million in the estimate of µ.

Government officials in Canberra have recently expressed concern regarding overruns on military contracts. These unplanned expenditures have been costing Australians millions of dollars every year. ...
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0answers
72 views

Normal Distribution - What should I study to understand these questions [on hold]

I got these questions from someone. I am not expecting anyone to 'solve' these, just need an idea of which specific topics I should read up on to help the concerned person understand, as I have been ...
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0answers
49 views

Understanding the solution of this integral [on hold]

The following integral represents an expected value of a geometric brownian motion for $S_T>K$: $$\int_{z^*} ...
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0answers
33 views

Division of Normal and Poisson Distribution

I am trying to understand the following: If $X$ is normally distributed and $Y$ is distributed according to the Poisson distribution, how to find out which distribution $Z=X/Y^2$ has? Is there any ...
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1answer
31 views

When will all the flowers blossom?

The title is not actually correct, but I chose appeal over correctness ;) I'd like to model a flower blossoming cycle, and these are the assumptions: 1) The instant $T$ in which each flower starts ...
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2answers
27 views

Finding Type I error

Suppose the sample size $n=16$ is drawn from a normal distribution with mean $\mu$ and standard deviation $\sigma = 4$. Consider the testing hypothesis $H_o:\mu = 0$ vs $H_a:\mu \ne 0$. Let the ...
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1answer
12 views

Distribution of inner product of normal random variable by a vector

Suppose, that we have a random vector $\mathbf{x} \sim \mathcal{N}(\mu,\Sigma)$. What is the distribution of $(a\cdot x)$, where $a$ is a real vector? It is known, that for a nonsingular real matrix ...
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0answers
16 views

Multivariate normal distribution problem

Consider three Gaussian variables $X_1,X_2,X_3$ with $\mathbb{E}[X_i]=0$ and $\mathbb{E}[X_iX_j]=\rho_{ij}$ for $i,j=1,2,3$. Then, three new variables are defined: $$ \left\{ \begin{array}{l1} ...
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1answer
13 views

Multivariate Normal cdf differentiation respect to dispersion

I am interesting in how to differentiate multivariate normal cdf respect to diagonal elements of covariance matrix (that is, I am interested only in variances). Problem similar to mine has been ...
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0answers
15 views

How to take partial derivative of a vector matrix vector multiplication?

I am trying to understand the mechanics of the below equations. I am especially confused about in 2.65 , how did the r.h.s which is a sum came from the gradient vector ? It would be great if someone ...
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2answers
630 views

Bivariate Normal Distributions

Let X and Y have a bivariate normal distribution with parameters μ1 =3, μ2 = 1, σ1^2 = 16, σ2^2 = 25, and ρ = 3/5 . Determine the following probabilities: (a) P(3 < Y < 8). (b) P(3 < Y < ...
3
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2answers
78 views

Boundedness of an integral of square function implying zero integral

Let $\alpha:\mathbb R\mapsto\mathbb R$ be the smooth function such that $$\int_{-\infty}^{\infty}[\alpha'(x)-x\alpha(x)]^2e^{-\frac{x^2}2}dx<\infty.$$ I wish to prove that ...
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0answers
14 views

Hypothesis test in Bayesian statistics

Let $X\sim N(\theta,1)$ and 5 independent observations $X=(4.9,5.6,5.1,4.6,3.6)$. The prior probability that $\theta=4.01$ is $0.5$. The remain values of $\theta$ are given the density of ...
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0answers
30 views

Integral (Tanh and Normal)

I am trying to evaluate the following: The expectation of the hyperbolic tangent of an arbitrary normal random variable. $\mathbb{E}[\mathrm{tanh}(\phi)]; \phi \sim N(\mu, \sigma^2)$ Equivalently: ...
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9 views

Compostion of tempered distribution and linear map.

While solving a particular problem about composition of tempered distributions and an affine transformation, I ended up having to prove the following for $u\in\mathscr{S}'$ and a linear transformation ...
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2answers
30 views
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1answer
4k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
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1answer
31 views

Probability that one normal (uncorrelated) variable is greater than another if the latter is positive

Assume that $X\sim N(0,\sigma_x^2)$, $Y\sim N(0,\sigma_y^2)$ and $X$ and $Y$ are uncorrelated. Can we solve analytically for $\mathbb P(X>Y |Y>0)$?
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1answer
22 views

Finding the cdf and pdf for $Z$, the standardization of $X$

Let $X$ be a normal random variable with parameters $\mu\in\mathbb R$ and $\sigma^2>0$. Find the cdf and pdf for $Z$, the standardization of $X$. What approach should I take on this? I initially ...
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1answer
29 views

Normal distribution pdf function returns value >1?

I am using the function scipy.stats.norm.pdf() in the following way: >>> scipy.stats.norm(scale=0.00026) >>> scipy.stats.norm.pdf(0.0005) 241.48 ...
2
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1answer
782 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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1answer
35 views

Use a $Z$ table to find $P(-1 < Z < 1)$. [closed]

Can someone help with these problems, please? Use Appendix Table III to determine the following probabilities for the standard normal random variable $Z$: (a) $P(-1 < Z < 1)$ (b) $P(-2 < Z ...
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0answers
20 views

Posterior of Normal with prior Cauchy

Let $X\sim N(\theta,1)$ and $\pi(\theta)\sim \mathrm{Cauchy}(0,1)$ find a 90% credible set for $\theta$ To find the credible set I need to find the distribution of $f(\theta\mid x)$, but ...
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1answer
40 views

Expected value of $X^{2n}$ where $X \sim N(0,1)$ [closed]

The question is: Show that if $X ∼ N(0, 1)$ has the standard normal distribution then $E[X^{2n}] = \frac{2n!}{2^{n}n!}$ Hint: compute the integral $\int_{-\infty}^{\infty} \frac{1}{\sqrt{2\pi}} ...
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56 views

Prediction Interval from Markov Chains

Thank you for taking the time to look at my question. Short, less involved question: How do you find Prediction Intervals with non-Gaussian residuals? Here is the situation: I have made a model that ...
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1answer
22 views

Using Moment Generating Function to prove Z is standard normal

Suppose $X_1,...,X_n$ is a random sample from a normal distribution with an unknown mean $\mu$ ,known standard deviation $\sigma$ and sample population $\bar{X}$. Show (using moment generating ...
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1answer
13 views

A specific question in conditional expectation with mixed discrete and continuous random variables

In my probability class I have just met this seemingly difficult question: Let $ \{X_n\}_{n=1}^{\infty}, \{Z_n\}_{n=1}^{\infty} $ be two i.i.d sequences of random variables such that we know $ ...
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2answers
84 views

Why $z(0.995)$ is $2.58$ and not $2.575$

Why some textbooks say that z(0.9950)=2.58, for instance "Statistics" by Murray R. Spiegel. Why don't they interpolate? If you look up in the z-table z(0.9949)= 2.57 and z(0.9951)=2.58 Thanks for all ...
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1answer
62 views

Help with this question from my textbook

Hello I've been battling with this particular question from my statistics textbook for hours. Can someone kindly help with this. Note: it is not an assignment question. I'm solving all questions in ...
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0answers
7 views

How to normalize sets of scores to have very similar histogram?

I have the output of several stochastic processes I need to combine into a single value. They have similar histogram curves, but not exactly the same. These curves are not perfectly Gaussian (see ...
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1answer
31 views

How to find the intersection of the normal CDF and the `y = x` line?

The normal distribution does not have a closed form cdf (per Wikipedia). Is there a way to find its intersection with the line x = y? Currently I use statistical ...
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1answer
32 views

How can I demonstrate that my data is sampled from a Gaussian process?

I have an experiment that, I believe, produces data with Gaussian noise. That is, any subset of my data points have a joint multivariate normal distribution with covariance K (i.e., they are sampled ...
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2answers
68 views

Distribution of a Gaussian variable with a normally distributed mean

Let $X\sim N(0,1)$ and $Y\sim(X,1)$, where $Y-X$ is independent of $X$. Then what is the PDF of $Y$? Specifically, I am interested in computing $P(Y<0\vert X>0)$. For those interested in ...
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1answer
6k views

Affine transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of ...
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2answers
4k views

How was the normal distribution derived?

Abraham de Moivre, when he came up with this formula, had to assure that the points of inflection were exactly one standard deviation away from the center, and so that it was bell-shaped, as well as ...