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

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0answers
23 views

Two random variables and finding the expectations

I have a question similar to this Link. The main difference is that now there exist two random variables, which are both normally distributed and independent to each other. There are two equations ...
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0answers
24 views

Maximum of a Gaussian random walk with non-identical steps

Consider a sequence of independent normal random variable $X_1,...,X_n$ with (negative) means $\mu_1,...,\mu_n$ and standard deviation $\sigma_1,...,\sigma_n$. Define \begin{equation} S_k = ...
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2answers
25 views

Errors and Residual

Why are errors independent but residuals dependent? As far i know the sum of the residuals within a random sample is necessarily zero, and thus the residuals are necessarily not independent. But also ...
1
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1answer
33 views

Normal Distribution Worded Problem

Standard deviation = 2.5 mL 98% of bottles must be between 998 mL and 1000mL Pr( 998 < x < 1000) = 0.98 This is a technology exam question, therefore to find the mean I used the method: ...
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0answers
12 views

Generate Correlated Normal and Log-Normal Random Variable

The standard approach for generating two normally distributed random variables some with correlation $\rho$ is explained here: Generate Correlated Normal Random Variables. Now let $X,Y$ be normally ...
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0answers
20 views

Derivative of the Inverse Cumulative Distribution Function for the Standard Normal Distribution

As the title says, I am trying to find the derivative of the inverse cumulative distribution function for the standard normal distribution. I have this figured out for one particular case, but there ...
0
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1answer
18 views

Standard deviation in normal distribution

A manufacturer uses a machine to make metal rods.The diameter of the rods follow a normal distribution with a mean of 1cm and a standard deviation of 0.02cm If the standard deviation of the diameters ...
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1answer
38 views

Convergence sequence of random variables

I have this problem about a sequence of normals. $(X_n)_{n\geq 0}$ is defined as $$X_{n+1}=aX_n+U_{n+1}$$ $X_0=0$, where $(U_n)_{n\geq1}$ is a sequence of i.i.d random variable normally distributed ...
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1answer
23 views

Normal distribution finding standard deviation [on hold]

A manufacturer uses a machine to make metal rods.The diameter of the rods follow a normal distribution with a mean of 1cm and a standard deviation of 0.02cm If the standard deviation of the diameters ...
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0answers
6 views

References to papers/books that uses a kernel to smooth a discrete distribution

Since a kernel, such as Gaussian, is often used to smooth out the distribution of discrete points in 1D, 2D or 3D, I believe there must be some study materials or research work that have used this, ...
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3answers
48 views

Transformation(?) of Random Variables

There are two independent Gaussian R.Vs: $U:N(-1,1)$ and $V:N(1,1)$ How do I go about finding the PDF of the following transformations? X = U+V T = (U+2V, U-2V) W = U (with 50% chance), V (with ...
2
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1answer
13 views

Distribution of a function of a normally distributed variable

Let's say you have a random variable $X$, which is normally distributed according to $X \sim \mathcal{N}(1,2)$. With $1$ being the mean and $2$ being the variance. Now let's say that there is another ...
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1answer
27 views

How to apply Gaussian kernel to smooth density of points on 2D (algorithmically)

I have a set of points on a 2D surface and need to build a heatmap. However, I also need to smooth out the density/distribution by applying some sort of kernel (Gaussian kernel, for example). I Know ...
2
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0answers
21 views

How to apply Gaussian kernel to smooth density of points on 2D (algorithmically) [duplicate]

I have a set of points on a 2D surface and need to build a heatmap. However, I also need to smooth out the density/distribution by applying some sort of kernel (Gaussian kernel, for example). I Know ...
0
votes
1answer
20 views

Expectation of an exponentiated quadratic form

Given a multivariate normal random $n\times 1$ vector $X \sim N(\mu,\Sigma)$, what is the expectation $$\mathbb{E}[exp(X^TAX+b^TX)]$$ where $A$ is a $n\times n$ matrix and $b$ is a n-dimensional ...
1
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1answer
25 views

if $X,Y$ i.i.d $\mathcal{N}(0,1)$, then $X+Y$ is independent of $X-Y$

I found on another thread* that if $(X+Y)$ is independent $(X-Y)$, and if $X,Y$ are i.i.d., then $X,Y$ are $\mathcal{N}(0,1)$ distributed. Is also the opposite true? Being $X,Y$ i.i.d ...
0
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1answer
40 views

Understanding edge correction with a 2nd order polynomial in Gaussian filter

I am trying to understand the following code from ImageJ: http://pastebin.com/tXfhNxqf The problem: When computing the gaussian kernel we use the gaussian function $$ f(x) = e^{-\dfrac{x^2}{2 ...
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0answers
35 views

Outliers in a Normal Distribution

Im doing AP Stat. in High School level. Here is a question i am stumped on because i feel like it is maybe a threory or law or something that i just never learned. However it DOES ask to show my ...
0
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0answers
39 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}} \mathrm ...
1
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1answer
30 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 ...
0
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1answer
38 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 ...
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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
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 ...
0
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1answer
69 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 ...
1
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1answer
36 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 ...
0
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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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0answers
8 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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0answers
46 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 ...
1
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1answer
28 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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1answer
22 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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2answers
34 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 ...
0
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2answers
42 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
34 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 ...
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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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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 ...
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0answers
6 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 ...
4
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1answer
54 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 ...
3
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3answers
98 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 ...
4
votes
1answer
164 views

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 ...
0
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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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0answers
58 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
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 ...
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 ...
0
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1answer
9 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 ...
0
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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 ...
0
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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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0answers
54 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 ...