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

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29 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
8 views

How to get Rayleigh distribution form Nakagami-Lognormal Distribution

Let $Z$ be a random variable that has Nakagami-Lognormal distribution given by ...
2
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0answers
16 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
29 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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0answers
32 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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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
64 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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0answers
68 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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1answer
37 views

Simulate two centered normal random variables with given variances and given covariance [on hold]

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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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
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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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} ...
1
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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 ...
-1
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0answers
42 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^*} ...
1
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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
29 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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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 ...
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2answers
29 views

Generating a random variable from a uniform random variable [on hold]

I have no idea how to go about doing this. Any help would be much appreciated.
1
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1answer
30 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 ...
0
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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 ...
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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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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
16 views
-2
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1answer
39 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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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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0answers
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 ...
1
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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 $ ...
2
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2answers
83 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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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
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
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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 ...
0
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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 ...
1
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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 ...
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0answers
17 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 ...
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1answer
12 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 ...
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0answers
16 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?
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0answers
11 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
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1answer
23 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 ...
1
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1answer
31 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
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2answers
56 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 ...
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0answers
39 views

Ratio of independant standard normal random variables.

I want to solve this question below. But I have no idea how to even start it. Any help would be appreciated.
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1answer
15 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
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2answers
39 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 ...
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1answer
18 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: ...
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0answers
14 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 ...
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2answers
13 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 $P(-4<Y<6)$.
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1answer
32 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 ...