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

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22 views

Integral of Normal Distribution with imaginary unit

Hi I need some help with the following integral. $$ \int_{-\infty}^{\infty} \operatorname{e}^{itx} \cdot \frac{1}{\sqrt{2\pi\sigma^2}} \cdot \operatorname{e}^{\frac{-(x - \mu)^2}{2\sigma^2}} dx$$ ...
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2answers
256 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 ...
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1answer
32 views

how to compute $E[e^{a^2/2}N^2]$, $N$ is $\mathcal{N}(0,1)$

I have to show that $E[e^{(a^2/2)N^2}]=E[e^{(aNN')}]$ and tell for which values of $a$ these quantities are finite. $N$ and $N'$ are independent $\mathcal{N}(0,1)$ random variables I computed the ...
2
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1answer
31 views

Bounding the norm of Gaussian random matrix

Suppose $A\in\mathbb R^{n\times m}$ is a random matrix with $n < m$, and each entry $A_{ij}$ follows i.i.d. Gaussian distribution $N(0,1/n)$. I want to know whether we can upper bound the spectral ...
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1answer
20 views

Conditional probability with a normal distribution

Given that Y and L are normally distributed, the expectation of L given Y is $\mu (Y)$ and the variance of L given Y is $\sigma ^2 (Y)$, why is the conditional probability $P(L > x| Y) = \Phi ...
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1answer
63 views

Transforming distributions

There is an economy, populated by a large number of agents. A first order condition common to all agents, is the following: $$E[\exp^{(1-\theta)\eta_i}(r-R+\eta_i)]=0$$ the index $i$ indicates the ...
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1answer
14 views

normality of data

Does the qqplot below suggest that the data is normally distributed? The fact that it's nearly perfectly linear is to me an indication of normality. However, the Anderson-Darling test for some reason ...
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1answer
129 views
+100

Solution of differential equation related to Normal density

Let $\phi:\mathbb{R}\mapsto\mathbb{R}$ be the standard normal density, $$\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}{2}}, \forall x\in\mathbb{R}.$$ Given $0<\sigma\le 1$. I wish to know whether there ...
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1answer
11 views

Discretization of normal distribution over a finite range

I only have data about the mean and standard deviation of a distribution over a finite discrete range (integers 1 to 5). How do i properly reconstruct the distribution (= a distribution that has the ...
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0answers
25 views

Closed-form term for this Gaussian expression

Related to this question, I have a random variable $X \sim N(\mu,\sigma)$. I am interested in $P[X \leq k]$ for $k=O(1)$, $\mu =o(1)$, $\sigma \sim \mu$. Is there some function $f$ such that ...
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1answer
34 views

Closed-form term for this expression

I have a normal Distribution $X \sim N(\mu, \sigma)$. Is there an easy way to give an asymptotic estimate with small error (I would prefer with relative error $\rightarrow 0$) for $P[X \geq k]$? We ...
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2answers
27 views

Normal distribution and algebra problem

Bags of cement are labeled $25 \operatorname{kg}$. The bags are filled by machine and the actual weights are normally distributed with mean 26.0 kg and standard deviation $0.50 \operatorname{kg}$. It ...
1
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1answer
186 views

$X$ and $Y$ i.i.d., $X+Y$ and $X-Y$ independent, $\mathbb{E}(X)=0 $and $\mathbb{E}(X^2)=1$. Show $X \sim N(0,1)$

$X$ and $Y$ are independent and identically distribued (i.i.d.), $X+Y$ and $X-Y$ are independent, $\mathbb{E}(X)=0$ and $\mathbb{E}(X^2)=1$. Show that $X\sim N(0,1)$. We should use characteristic ...
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1answer
79 views

For a normally distributed random variable, find a value from given tail probability

Problem Let $ X \sim N(65,20) $. Find correct to $3$ Decimal Place the value of $x$ such that $Pr(X>x) = 0.43$. Progress I've gotten to $\frac {x-65}{2(5)^{1/2}} =0.1764$ and hence $x = 67.789$? ...
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0answers
41 views

Vector distribution after Girsanov transform

Let $X$ be a gaussian vector under $P$ and $U$ a variable such that the vector $(X,U)$ is gaussian. $dQ = Y dP$ with $Y = e^{(U −E_p(U) − 1/2 var_p[U])}$. I have to show that $X$ is gaussian under ...
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0answers
7 views

Comparing normal distributions using a two sample Kolmogorov-Smirnov test

I have used a two sample Kolmogorov-Smirnov test to compare the distributions of two sets of data. I know that the K-S test is a non parametric test, however the distributions of data I'm comparing ...
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1answer
22 views

convergence to standard brownian motion

Could you help me with the following: I have that $$T(x):=\frac{X(nx)-E[X(nx)]}{\sqrt{n}} \xrightarrow{d} N(0, \frac{x^k}{k})$$ for each fixed $x>0$, where we also have that $\frac{X(nx)}{t}$ is ...
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1answer
27 views

Probability that $f(x,y,z)>0$ given the variables follow normal distribution

Assuming that variables $x,y$ and $z$ follow the Gaussian distribution with $\mu_x=\mu_y=\mu_z=1000000$ and $\sigma_x=\sigma_y=\sigma_z=200000$, what is the probability that $$f(x,y,z) = ...
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2answers
33 views

Why we consider log likelihood instead of Likelihood in Gaussian Distribution

I am reading Gaussian Distribution from a machine learning book. It states that - We shall determine values for the unknown parameters mu and sigma^2 in the gaussian by maximizing the ...
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1answer
154 views

Probability (Statistics) [closed]

The internal sales group of a company has full-time employees on the phone calling prospective customers. Based on historical information, each call only has a 5% probability of being successful. ...
1
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1answer
20 views

interpret the histogram (generated in excel)

I have generated and attached the histogram here for reference. On X-axis it's time in hour Considering, mean=7.52, SD=1.71, upper bound =7.76, lower bound=7.28, confidence interval=96% - What is ...
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1answer
40 views

Interval Estimation when $\overline{Y}$ and $S$ is unknown.

Question: A random sample of size $n=9$ is drawn from a normal distribution with $\mu=27.6$. Within what interval $(-a,+a)$ can we expect to find $\frac{\overline{Y}-27.6}{S/\sqrt{9}}$ $80$% of the ...
8
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1answer
281 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
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2answers
41 views

probability of normally distributed variable being greater then another normally distributed variable

i have seen this question being addressed around, but I have problem with deriving the proof. Namely, if we have two normally distributed variables, $x$ and $y$, with their distributions given as ...
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0answers
33 views

Determining $\sigma$ given mean and proportion of a Normal distribution?

The marks of a random sample of students with mean $\mu$ and standard deviation $\sigma$ showed that 15.87% scored higher than 70. The distribution of the marks is Normal with mean $50$ standard ...
5
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1answer
67 views

An interesting inequality about the cdf of the normal distribution

When approaching this other question I came out with the inequality: $$\frac{1}{4+x^2}e^{-x^2/2} \leq\Phi(x)\Phi(-x)\leq \frac{1}{4}e^{-x^2/2},\tag{1}$$ where $\Phi(x)$ is the cdf of the standard ...
4
votes
1answer
49 views

Abramowitz and Stegun approximation for cumulative normal distribution

(Note: I know this looks like a programming question, but I'm OK with the programming part and just want to understand the mathematics.) I found a bit of code to calculate the integral of the normal ...
0
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0answers
6 views

Show that $\partial_i\psi(x;\Sigma^{-1},\mu) = -\Sigma^{-1}_{ii}$

Let (1) $\psi(x;\Sigma^{-1},\mu) = \Sigma^{-1}(x-\mu)$. Now, (2) $\partial_i\psi(x;\Sigma^{-1},\mu) = -\Sigma^{-1}_{ii}$ How do you arrive at (2)? See: ...
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0answers
5 views

covariance and correlation of two three dimentional gaussian distributions

Lets say we have 'n' three dimensional dimensional gaussian distributions with a '3' dimensional mean vector and 3 x 3 non-diagonal covariance matrix. How can I check if they are correlated? Is ...
3
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3answers
96 views

Mean and Variance of “Piecewise” Normal Distribution

Note - I put piecewise in quotes because I don't think it's the right term to use (I can't figure out what to call it). I am building a program to model the load that a user places on a server. The ...
4
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1answer
42 views

How to show that $\int\limits_{-\infty}^{+\infty}(n-1)\Phi(x)^{n-2}\phi(x)^2dx$? decreases in $n$?

I was working on a research project that involves taking the integral of $$(n-1)\int\limits_{-\infty}^{+\infty} \Phi\left(x\right)^{n-2}\phi\left(x\right)^2dx,$$ where $\Phi(.)$ is the CDF for ...
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3answers
309 views

Scaling the normal distribution?

I might just be slow (or too drunk), but I'm seeing a conflict in the equations for adding two normals and scaling a normal. According to page 2 of this, if $X_1 \sim N(\mu_1,\sigma_1^2)$ and $X_2 ...
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1answer
214 views

comparing two normally distributed random variables

In a large corporation, people over age thirty have an annual income whose distribution can be approximated by a normal distribution with mean 60,000 and standard deviation 10,000. The incomes of ...
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1answer
39 views

Determine the target weight so that no more than 5% of boxes with normal weight distribution contain less than 500 g [closed]

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. Suppose a law states that no more than 5% ...
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1answer
258 views

Generalized chi distribution

Let $v\in\mathbb{R}^n$ follow a multivariate Gaussian$(0,I)$ distribution, and $M\in\mathbb{R}^{n\times n}$ a matrix. Has the distribution of the Euclidean norm $\|Mv\|$ been studied? I know that its ...
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0answers
57 views

Approximation for the convolution of normal and lognormal distributions

$$X \sim \ln\mathcal{N}(\mu_X,\,\sigma_X)$$ $$Y \sim \mathcal{N}(0,\,1)$$ $$Z = X + Y$$ I want to find the probability density functions and cumulative distribution functions of $Z$. As the below is ...
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1answer
41 views

The probability that a joint distribution is less than a certain value, given the correlation coefficient.

For this problem, we are told that $X$ and $Y$ are jointly normally distributed variables, both being standard normal. We're given their correlation coefficient. So, how do I get from there to finding ...
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1answer
16 views

Solving a statistics equation

Suppose $X$ is a random variable which follows a Poisson distribution, such that, for some positive integer $m$, $$X \sim Po(0.01m)$$ Find the least value of $m$ such that $$P(X \ge 1) > 0.9$$ ...
0
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1answer
25 views

generate random number from normal distribution

Can any one explain in which range I am going to get random numbers, if I was said generate random number from normal distribution with mean=50 and std_dev=25, what does it exactly means..I tried to ...
1
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1answer
7k views

Find a Probability of a Normally Distributed Random Sample

Please help me figure out how to do this problem. I need to be able to understand how to solve problems like this. Thanks times a million! Problem: An employer is interested in the commute times for ...
0
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0answers
9 views

Variance reduction factor using gaussian filtering

I am currently trying to find the variance reduction ratio using gaussian filtering. For a simpler filter (as mean filtering for example), I am able to calculate it easily to find the well known ...
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1answer
58 views

Getting a p-value from a histogram?

A hypothetical HIV vaccine trial involving 20,000 participants—10,000 in the vaccine group and 10,000 in the placebo group—had the following results: 6.3 infections per 1000 in the vaccine group and ...
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1answer
35 views

Confidence Interval for Regression Coefficient ,$\beta$

In the book 'Applied regression Analysis' by Draper/Smith, it is written that : Obtain individual $100(1-\alpha)\%$ confidence interval for the various parameters separately from the formula ...
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1answer
97 views

Find the warranty period such that the battery is replaced under warranty 0.5% of the time

Problem The mean life of a Chevy Volt battery (normally distributed) is $1000$ hours and the standard deviation is $100$. How many hours should GM warranty the battery for so that it has to replace ...
0
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1answer
227 views

Mixture Gaussian distribution quantiles

Let $f_1(x), \dots, f_n(x)$ be Gaussian density functions with different parameters, and $w_1, \dots, w_n$ be real numbers that sum-up to unity. Now the function $g(x) = \sum_i w_i f_i(x)$ is also a ...
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1answer
8 views

Modelling a normal-like single-ended random variable

I am trying to model a of (normal-distribution-like) discrete random variable using the normal distribution. This is what I understand so far: First, I approximate the mean of the normal ...
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1answer
17 views

Regression when the variance of the residuals depends on the independent variable

When the residuals follow a normal distribution, the most likely function that fits the data is found using least squares. In that case: $y = f(x_i) + r_i, \quad r\sim\mathcal{N}(0, \sigma^2)$ ...
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1answer
241 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 ...
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0answers
50 views

Compound Gaussian distribution

Let $\mathbf{a},\mathbf{b}\sim \mathcal{N}(\mathbf{0},\sigma^2\mathbb{I})$ and let $A$ be the circulant matrix defined to have $\mathbf{a}$ as its first column. I'm trying to study the behaviour of ...
2
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1answer
332 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 ...