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

learn more… | top users | synonyms

1
vote
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
vote
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
vote
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
vote
2answers
83 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
votes
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
votes
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
votes
1answer
171 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
328 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
221 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
votes
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
votes
1answer
285 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
vote
0answers
213 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
vote
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
65 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
votes
0answers
51 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
votes
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
votes
1answer
460 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
votes
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
33 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
vote
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
votes
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
vote
1answer
73 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
votes
1answer
36 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
votes
2answers
53 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
vote
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
votes
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 ...
0
votes
2answers
48 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
votes
2answers
123 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
votes
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
vote
1answer
81 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
vote
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
43 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
votes
3answers
64 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
votes
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. ...
0
votes
0answers
23 views

User of a System

Given a system with n users and each user will only use the system once (for an hour) during a year. The user will only access the system during business hours (so ...
3
votes
2answers
68 views

$\frac{1}{\sqrt{2\pi}}\int_\frac {1}{2}^0\exp(-x^2/2)dx$

How do we analytically evaluate $J=\frac{1}{\sqrt{2\pi}}\int_\frac {-1}{2}^0\exp(-x^2/2)dx$? This is what I tried: $$ J^2=\frac{1}{{2\pi}}\int_\frac {-1}{2}^0\int_\frac {-1}{2}^0\exp(-(x^2+y^2)/2)dxdy ...
1
vote
0answers
92 views

Probability:questions on characteristic functions

A well-known example to show that two random variables whose marginal distributions are normal, do not need necessarily be jointly normal is achieved by letting $X, Y $ have the following joint ...
1
vote
1answer
74 views

How to fill in these steps to evaluate this Gaussian integral?

As a part of a much bigger problem, I came across this integral $$\int_{-\infty}^{\infty}\ln(|x|)\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2}dx$$ which represents ...
0
votes
0answers
18 views

Combining independent Gaussian probabilities

I am using three Gaussian distributions with which I generate random numbers to represent many candidate xyz points. I use some selection criteria (details not particularly relevant) to decide on ...
2
votes
0answers
68 views

On integration of a Gaussian-like function over a region $g(\mathbf{x})\leq 1$

Let $X$ be a random variable which follows an $n$-dimensional Gaussian distribution with mean vector $\mu\in\mathbb{R}^n$ and covariance matrix the symmetric positive definite $n\times n$ matrix ...
2
votes
1answer
39 views

Marginalization of a paramter in Gaussian

If $\theta \sim N(\mu,\sigma_o^2)$ and $\mu \sim N(0, \sigma_1^2)$ what is the marginalized $P(\theta)$. Is it $N(0,\sigma_o^2+\sigma_1^2)$?
3
votes
1answer
207 views

Is there an analytical solution to Gaussian integral $\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx$?

I wonder if there is an analytical solution to $$\int_{-\infty}^{\infty} \frac{e^{-x^2}}{(x+a)^2+b} dx,$$ where $a, b>0$. I know, of course, that the antiderivative of the fraction is a version ...
0
votes
1answer
70 views

integral with pdf of a gaussian

$$ I = \int_{0}^{\infty} x \phi(x) dx $$ where $\phi(x)$ is the pdf of a normal distribution. Here I read that: If $X = \mu + \sigma U$ with $U$ a std normal, $$ I = E[\mu + \sigma U; mu + \sigma ...
1
vote
1answer
23 views

Will statistical analysis of transformed data hold for the original one?

I have a data with distribution like chisq-squared one. But ANOVA and t-test need the data to be normal distributed. So I want to do the Box-cox transformation to the data, but my concern is will the ...
-1
votes
1answer
36 views

Normal distribution Z score

Problem: The observed error "E" in a series of measurements is normally distributed with mean of 0. Approximately 2% of error are -10 or less. Approximately what fraction of the measurements have ...
2
votes
1answer
35 views

Show that $d^T Z\sim N(d^T\mu, d^TVd)$ [duplicate]

Consider $Z=(Z_1,\ldots,Z_n)^T\sim N(\mu,V)$ with $\mu=(\mu_1,\ldots,\mu_n)^T$ and $V=\text{Cov}(Z)$. Show that for $d\in\mathbb{R}^n$ it is $$ d^TZ\sim N(d^T\mu,d^TVd). $$ For me it ...
2
votes
1answer
34 views

Gaussian prior favors values closest to zero?

I am reading an article on Bayesian Logistic Regression, where they're using Logistic Regression, imposing a Gaussian prior (with mean = 0) on its parameters. They state that a Gaussian prior favors ...
0
votes
1answer
66 views

Given $f_X$. Integrate $\int_0^\infty \log_2 (x+1) f_X \, dx$.

Say $Y=Log_2[1+x]=g(X)$ and $f_X = \frac{e^{-\frac{(\mu -\log (x))^2}{2 \sigma ^2}}}{\sqrt{2 \pi } x \sigma }$ is Log-normal density function: [Wiki] Find E[Y]? Since $E[Y] = \int_0^\infty y f_Y \ ...