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

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1answer
131 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
44 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
44 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
48 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
1k 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 ...
1
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1answer
66 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
39 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 ...
0
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1answer
69 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 ...
0
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1answer
132 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
98 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 ...
2
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1answer
43 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 ...
0
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2answers
45 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
37 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
18 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 ...
1
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1answer
36 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) = ...
1
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1answer
46 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 ...
1
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2answers
89 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
55 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 ...
0
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0answers
13 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: ...
6
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1answer
182 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
357 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
votes
3answers
229 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
50 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 ...
9
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1answer
287 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 ...
1
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0answers
226 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 ...
1
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1answer
18 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
votes
1answer
66 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
56 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
17 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
677 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
27 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: $y = f(x_i) + r_i, \quad r\sim\mathcal{N}(0, \sigma^2)$ ...
2
votes
1answer
34 views

Minimum number of samples to take so that proportion of smokers in sample is within a certain threshold?

What is the minimum number of random samples that should be taken so that with probability at least 0.95, the proportion of smokers in the sample will not differ from the unknown population of smokers ...
1
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0answers
17 views

Finding posterior of normal distributions and logistic regression.

$P(w_0 | x) = \frac{1}{1 + e^{-log\frac{P(x|w_0)}{P(x|w_1)}-log\frac{P(w_0)}{P(w_1)}}}$ Note: x = $[x_1, \dots, x_d]^T$; a $d$ dimensional vector. $w$ can take on one of two values: $w_0$ or $w_1$. ...
-1
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1answer
29 views

Distribution combinations [closed]

How many ways can $25$ identical pencils be distributed between two people? Each pencil must be given out. a) Each person must have at least $5$ pencils. b) Each person must have at least $7$ ...
1
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1answer
79 views

Sum of dependent normal random variables

Let ${\bf X} =(X_1,\ldots,X_n)'$ be a vector of random variables that may be dependent and let ${\bf a}=(a_1,\ldots,a_n)'$ and ${\bf b}=(b_1,\ldots,b_n)'$ be nonrandom vectors with $a_i \neq 0$ and ...
0
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1answer
37 views

Proving some properties about the expected first order statistic (maximum) with respect to sample size.

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as ...
3
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2answers
54 views

What is the reason for the one-half in the normal pdf's gaussian (i.e. : why $\exp(-x^{2}/2)$ instead of $\exp(-x^{2})$ )

It doesn't seem to relate to normalization, as the normalizing constant adapts to every possible "upstairs formulation", and in the standard case is $\displaystyle\frac{1}{\sqrt{2\pi}}$. Does it ...
1
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1answer
75 views

Explain why $\big(\int_{-\infty}^{\infty}e^{-z^2/2}dz \big)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 + u^2)/2}dzdu$

I came across the following when studying a proof related to the normal distribution: $$\left(\int_{-\infty}^{\infty}e^{-z^2/2}\ dz \right)^2 = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(z^2 ...
3
votes
4answers
261 views

$\int_{0}^{\infty}xe^{-x^2/2}dx= 1$?

$X \sim N(0, 1)$ $$E(|X|) = \frac1{\sqrt{2\pi}}\int_{-\infty}^{\infty}|x|e^{-x^2/2}dx= \frac{2}{\sqrt{2\pi}}\int_{0}^{\infty}xe^{-x^2/2}dx=\sqrt{\frac{2}{\pi}}$$ I don't understand how the last ...
2
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1answer
24 views

Surjectiveness of standard-normal c.d.f. [closed]

Let $\phi:\mathbb R \to (0,1)$ be a function defined as $\phi(y)=\int_{-\infty}^y\dfrac{1}{\sqrt{2\pi}}e^{-\dfrac {x^2}{2}}dx , \forall y\in \mathbb R$ , then is it true that $\phi$ is surjective ? If ...
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2answers
53 views

Normal Distribution,standard deviation and probability question.

According a study, the duration of a match in World Cup is approximate normally distributed with the mean 111 minutes and standard deviation 5 minutes (including the break between the halves). ...
2
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2answers
127 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
0
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2answers
35 views

Area under Normal Distribution Curve

What is the formula that determines the Z-score table? More specifically, what formula can be used the equate the area underneath the normal distribution curve, without using the table?
1
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1answer
87 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 ...
1
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1answer
60 views

Space of Gaussian Functions is Closed in $L^2$

Let $\Omega, \mu$ be a probability space. A measurable function $f: \Omega \rightarrow \mathbb{R}$ is called Gaussian if $$\mu (f^{-1}(A))=\frac{1}{\sigma \sqrt{2\pi }}\int_Ae^{-x^2/2\sigma ^2} dx$$ ...
2
votes
1answer
57 views

Generating a nonrandom sequence which has a normal distributed density

I need to create an algorithm in a computer program (Fortran90) which generates a sequence of $n$ (between $10$ and $10^6$) numbers $z$ that follow a normal distribution. Restrictions: Has to ...
0
votes
1answer
44 views

Mantel-Haenszel $\chi_1^2$ statistic

I was doing a particular example from the book Epidemiologic Research by Kleinbaum(example 15.6) and didn't understood some basic statistical aspect. ...
0
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3answers
65 views

Question about normal approximation and variance

This isn't so much a question about getting a right answer as much as it's about understanding a mathematical concept, but I will give you the problem that spawned it: An analysis of data shows that ...
0
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1answer
40 views

Plotting Normal Distribution using Excel

I was trying to experiment some stuff (scaling issues and hypothesis testing) with normal distribution. While doing so, I found out that : NORM.S.DIST(0, FALSE), which takes Z-value, returns prob. ...