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

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2
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0answers
29 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 ...
1
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0answers
36 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 ...
0
votes
0answers
41 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 ...
0
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0answers
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. ...
1
vote
1answer
55 views

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 N\left(0,\dfrac{(\sigma^{2})^{3}}{3}\right)\qquad\...
1
vote
2answers
31 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 ...
0
votes
1answer
17 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 $...
0
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0answers
16 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 ...
1
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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} Y_1 ...
1
vote
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 ...
1
vote
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 $g(\...
1
vote
0answers
34 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: $...
0
votes
0answers
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 ...
-2
votes
2answers
33 views

Generating a random variable from a uniform random variable [closed]

I have no idea how to go about doing this. Any help would be much appreciated.
1
vote
1answer
32 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 $\mathbb P(X>Y |Y>0)$?
0
votes
1answer
23 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 ...
0
votes
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 ...
0
votes
1answer
15 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 ...
1
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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 $$f(\...
0
votes
1answer
17 views
-2
votes
1answer
44 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}} e^{-...
0
votes
1answer
30 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 ...
1
vote
0answers
58 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 ...
1
vote
1answer
17 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 $ ...
2
votes
2answers
93 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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
1answer
34 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 ...
0
votes
1answer
33 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 ...
1
vote
1answer
17 views

How to find $\Phi^{-1}(\beta)$

I need to find $\Phi^{-1}(\beta)$ when $\beta=0.1$ (or any number but for example) but I'm not quite sure how to find it using the normal table inversely like this. I've tried googling and looking ...
0
votes
0answers
18 views

What does this question mean? The Wald test

Looking at a past paper without soltuions, I am unclear of what is being asked. Context $x_1...x_n$ denotes a random sample from a normal distribution $N(\mu,\theta)$. After I've obtained the ...
0
votes
1answer
13 views

Bivariate normal covariance

I'm trying to prove that the covariance of a bivariate normal distribution, $Cov(X,Y)$, is equal to $\rho\sigma_X\sigma_Y$. I'm getting stuck. The approach I have to use is to apply a change of ...
0
votes
0answers
21 views

The quotient of two chi distributions

The quotient distribution of two chi-squared distributions is F-distribution. What would be the quotient distribution of two chi distributions? Is there a general distribution for this?
0
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0answers
15 views

How to obtain gaussian(normal) distribution

I heard that Gaussian distribution(Normal distribution) is obtained by maximum entropy theorem. Using lagrange mutilplier, Gaussian distribution is easily obtained. However, it's too hard for me. ...
0
votes
1answer
25 views

Pairwise independence implies independence for normal random variables

I'm reading a book on Brownian Motion. In the proof of the existence of such random function (Wiener, 1923), the following is stated: Indeed, all increments $B(d)-B(d-2^{-n})$, for $d\in \mathcal{...
1
vote
1answer
32 views

Basic questions concerning sample means and distributions

I have the following questions: Is The value of the sample mean always the population mean $\mu$, in any sample? I am confused about whether or not it is. Is the sampling distribution of the sample ...
2
votes
2answers
59 views

Showing two random variables independent despite seemingly looking dependent

I just met this in probability and it got me completely stumped: We define an i.i.d sequence of normally distributed random variables $ \{ X_n \}_{n=1}^{\infty} $ such that $ X_n \sim \mathcal{N}...
0
votes
1answer
27 views

How to estimate the max of a population using the normal distribution equation on a small sample

I recently watched a documentary on Mathematics. In the show they managed to estimate the weight of the largest fish that the fisherman was likely to of ever caught in his career just by analysing one ...
0
votes
2answers
43 views

A worker is told that only $5\%$ of all workers make a higher wage. If the wages are normally distributed, what is the average hourly wage?

So let's say a worker earns $\$16$ per hour at a plant and is told that only $5\%$ of all workers make a higher wage. If the wages are normally distributed with standard deviation of $\$5$ per hour ...
0
votes
1answer
25 views

Box-Muller method to Polar Marsaglia scheme

I have just learned the Box-Muller method for generating normal random values. My notes then consider the Polar Marsaglia method, which is more efficient than Box-Muller. In Box-Muller: $$X=\sqrt{-2\...
0
votes
1answer
18 views

Error function relation to the normal cumulative distribution function

A CDF for a normal standard is the following: $$N(x) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^x e^{-\phi^2/2} d\phi$$ I have the following relation in my notes which I am not very sure how they ...
1
vote
2answers
18 views

Maximum and minimum value of $P(-4<Y<6)$, where Y has normal distribution with standard deviation 2 and the mean unknown.

Let Y be a random variable has normal distribution with standard deviation 2 and mean is unknown. Find the maximum and minimum value of $P(-4<Y<6)$.
0
votes
1answer
36 views

T-Distribution, Normal Distribution, and Confidence Intervals

In my probability class we were given the following problem: Suppose you take a sample of your friends and measure their heights. You calculate the sample mean to be 5 feet tall and the sample ...
0
votes
1answer
42 views

Shape of chi-square distribution df=1

I am trying to understand, intuitively, the shape of the chi-square distribution with 1 degree of freedom. Let $X$ be a random variable whose distribution is given by the standard normal distribution....
3
votes
1answer
32 views

Probability that a Wiener process is negative at 2 given that it was positive at 1

Let $W_t$ be a standard Wiener process, i.e., with $W_0=0$. If $W_1>0$, what is the probability that $W_2<0$? This is my attempt: we want to determine the conditional probability $$\mathbb P(...
0
votes
0answers
15 views

Relationship between CDF of two normal distributed variables?

If F(X) is cumulative density function of normal distribution of mean 0 and standard deviation of 1, then what is its relationship with F(kX). Here k is constant. Can we express it as g(k)*F(X)?
1
vote
2answers
14 views

Confidence Itervals; $Z_{\alpha}$ & $Z_{\alpha/2}$

I'm confused about what exactly $Z_{\alpha}$ is, does there exist a formula for it in terms of $\alpha$? IF so, is there also one for $Z_{\alpha/2}$?
0
votes
1answer
14 views

Gaussian expected inner product with respect to fixed matrix

Suppose $R\in\mathbb{R}^{p\times p}$ is a a fixed matrix (it can be asymmetric, non-positive definite, and so on). Then, I would like to find a formula for the expectation $$\mathbb{E}_{x\sim N_p(0,V)...
0
votes
1answer
30 views

Prove $\lim_{\Delta t \to 0} \frac{2}{\Delta t} \left(1-\Phi\left(\frac{\epsilon}{\sqrt{\Delta t}}\right)\right) = 0$

Given $$\lim_{\Delta t \to 0} \frac{2}{\Delta t} \left(1-\Phi\left(\frac{\epsilon}{\sqrt{\Delta t}}\right)\right) $$ with $\Phi$ standard normal CDF, how can I prove the limit to be equal to $0$? ...
1
vote
2answers
32 views

get data to draw a gauss curve

I would like to know how to get some data from a normal distribution to draw its gauss curve. I have the standard deviation, the average and the x, but I don't know how to get some points to draw the ...