The normal, or Gaussian, distribution has a density function that is a symmetrical bell-shaped curve. It is often used as a reference against which other distributions are compared.

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11 views

What do the remaining columns (after column 2) in a Z table mean

In the Z table , I can understand the first 2 columns. The first column is the Z value, the second column is Prob(X<=Z). But what do the remaining columns mean?
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8 views

Analyzing time to event variable that has an upper limit cutoff

Setup: 4 different interventions, outcome metric: time for subject to respond. So lets say that for three of the groups of subjects they responded within 5 to 20 seconds. However, in one group, the ...
4
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2answers
88 views

Inference on $P\left(\left.\sum_{i=1}^{N}X_{i}\ \right|\ \sum_{i=1}^{N}X_{i}^{2}\right)$ when $X_{i}\sim\mathcal{N}\left(0,1\right)$?

Let: $$X_{i}\overset{i.i.d}{\sim}\mathcal{N}\left(0,1\right)$$ Hence: $$\sum_{i=1}^{N}X_{i}\sim\mathcal{N}\left(0,N\right)$$ and $$\sum_{i=1}^{N}X_{i}^{2}\sim\chi^{2}\left(N\right)$$ What can ...
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0answers
30 views

Under what circumstances will the log of a variable be normal, given that the variable is not normal

Let us say there is a variable that is not normally distributed. Under what circumstances will the natural logarithm of the variable be normally distributed? I have seen many articles and papers ...
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10 views

Axes having different scales on a q q plot

I was trying to learn linear regression in SAS using this example When I came to the part about the q-q plot: ...
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0answers
12 views

Comparing normalized and partially normalized quadratic form of normal variables

Suppose $X=(X_1,X_2,\ldots,X_n)' \sim \mathcal{N}(0,\Omega)$, where $\Omega$ is the variance-covariance matrix of dimension $n\times n$ for the vector $X$. Let $a=X'\Omega^{-1}X$. It is known that ...
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0answers
5 views

Objective evaluation of subjective data points and normalization

I want to write an access database that takes in inputs from instructors ranking a student on their performance in a handful of categories. (1 - below average, 2 - average, 3 - above average). They ...
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0answers
15 views

Efficient computation of a (truncated) distribution's parameters given the moments (and back)

I have a probability distribution of the form: $$p(\vec x) = Z^{-1} \delta\left(\sum_i x_i\right)\exp \left[ \sum_i (\phi_i x_i + \psi_i x_i^2) \right]$$ for $\vec a \le \vec x \le \vec b$, and ...
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0answers
16 views

Applying parametric tests on non-parametric data

I'm doing a research and I have some concerns, and I'd appreciate your kind assistance on them. Basically, I'm designing an instrument to measure something (a single dependent variable), and I'm ...
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0answers
16 views

Combining two conditional probability distributions; what is the variance?

Say I have two conditional probability distributions: ($Y_1$ | $Z_1$,$Z_2$) ~ N ($\dfrac{a_1B_2Z_2}{1-a_1a_2}$ , $\dfrac{a_1^2+1}{(1-a_1a_2)^2}$)    and ($Y_2$ | $Z_1$,$Z_2$) ~ N ...
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2answers
32 views

How close does a distribution have to be to normal, in order for predictions to be accurate?

I know a Kolmogorov-Smirnov test will tell me if a sample distribution belongs to a normal distribution or not, with a certain probability (correct me if I am wrong?). I performed a KS test on my ...
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0answers
57 views

Log-likelihood proof and AIC hypothesis

First of all, statistics is just not my thing ... yet (I hope!) I'm having a hard time finding out the log-likelihood equation: Given $Y \rightarrow \mathcal{N}(\mu_1,\sigma_1)$ (observation) and ...
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0answers
14 views

Without using random function, how can i generate multivariate normal distribution? [closed]

mu = [2,3]; sigma = [1,1.5;1.5,3]; mvnrnd(mu,sigma,100); this function in matlab gives 100 samples which is normally distributed with mean 'mu' and covariance matrix 'sigma'.I don't want to use random ...
1
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1answer
53 views

Multi Armed Bandit for Continuous Rewards - Extended Question

This question is an extension to A continuous generalization of the binary bandit The Multi-Armed Bandit (MAB) Problem in general is described here: https://en.wikipedia.org/wiki/Multi-armed_bandit ...
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1answer
45 views

constant $times$ distribution

I know that if $U\sim\chi^2(k)$ then $aU\sim \Gamma(k/2,2a)$ for $a>0$. But i read about the estimator and its distribution $$\hat{\sigma}_k^2=\frac{1}{2k}\sum_{i=1}^k ...
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1answer
32 views

Generate random number with normal distribution?

I encountered this question, where given it is a normal distribution, how do we go about it to generate a series of random numbers beside Monte Carlo? The clue given was using exponential function.
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1answer
37 views

Covariance Matrix with all equal entries

by training a Gaussian Process Regression Model I'm finding the weird result where the resulting covariance matrix has all the entries equal between each others. I'm using a Gaussian kernel with ...
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1answer
31 views

Can we calculate Z-score for any distribution?

Is z-score only confined to normal distribution or can it be used for any distribution
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0answers
9 views

Comparing z scores to Binary data

I have a set of quantitative data for which i have calculated the z scores. I also have a set of qualitative data in which parameters have been assigned scores of 0 or 1 based on expert opinion My ...
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2answers
34 views

How to estimate 100% confidence interval aka. what is the Z value of standard normal distribution at probability of 100%?

Thinking of the various tests and parameter estimates we perform with 99% confindence interval based on assumption of "normal distribution of errors" I asked myself a question what would be the 100% ...
3
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0answers
56 views

Test for difference in proportions with small probabilities

I want to perform a test for differences between 2 binomial populations. The probabilities are small (usually less than 10 %). I can define the successes as "rare events". I think that the z-test ...
2
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0answers
52 views

Distribution of geometric mean of normally distributed independent random variables [closed]

Suppose we have $X(i)$ for $i=1,...,n$ normally distributed independent random variables with the same known $\mu$ and $\sigma$ for all: $$X(i) \tilde{} N(\mu,\sigma)$$ Suppose that we take the ...
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0answers
34 views

PDF of multivariable function from known distribution of components [duplicate]

How can I determine the pdf of the following function: $$z(x,y) = \sqrt{ax^2 + by^2}$$ given the constants $a,b$, the means $\mu_{x},\mu_{y}$ and variances $\sigma^2_{x},\sigma^2_{y}$ of the ...
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1answer
49 views

Deegrees of freedom of Student's distribution

I'm trying to figure out the distribution of this statistic: $$S=\frac{\frac{\overline{X}-\mu_0}{\sigma / \sqrt{n}}}{\sqrt{\hat{\sigma}^2/\sigma^2}}$$ Where: $\overline{X}=\frac{1}{n} \sum_{i=1}^n ...
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0answers
16 views

Conjecturing asymptotic normality for a sum of dependent random variables

I am hunting for the asymptotic distribution of a scaled partial sum of pair-wise equi-correlated, identically distributed continuous random variables $$W =n^{-\delta}\sum_{i=1}^nY_i(n),\;\; ...
2
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0answers
47 views

How to invert a sparse covariance matrix for spatial data on a grid?

Say we have some gaussian random variables that can be indexed on a grid. A convolution was applied to this grid, so now there is covariance between the grid points. The covariance is given by (see ...
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2answers
270 views

Which probability distribution fits my data?

I have generated a dataset (available here) for which I try to find out the best fitting probability distribution. I first generated uniformly distributed random directions and then calculated the ...
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0answers
10 views

Truncated/censored AR1 normal likelihood

I have a model for some data that I am analysing which is of the form: $W^*_t = \rho W^*_{t-1} + \epsilon_t$ Where $\epsilon_t\sim N(0,\sigma^2)$. $W^*_t$ is a latent (hidden) process, which ...
0
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0answers
16 views

Sending a variable through multiple distributions

Say you have 5 people and 100 lbs of food. Person 1 (P1) gets to the food first, takes some food, and passes it to P2. P2 takes some of the food and passes it to P3, and on to P4. P5 gets whatever is ...
4
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2answers
82 views

Expected magnitude of a vector from a multivariate normal

What is the expected magnitude, i.e. euclidean distance from the origin, of a vector drawn from a p-dimensional spherical normal $\mathcal{N}_p(\mu,\Sigma)$ with $\mu=\vec{0}$ and $\Sigma=\sigma^2 I$, ...
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48 views

How to estimate mean and variance of censored normal?

Supposing I have data which I know is normally distributed, but because the recording process is right censored, how do I estimate the parameters of the distribution?
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5answers
299 views

Standardizing a Standard normal Variable

If I standardize a standard normal random variate , will it be still standard normal ? That is, if $X\sim N(0,1)$ , then can I do $$X^*=\frac{x-\bar x}{sd(x)}$$ ? and will $X^*\sim N(0,1)$ ? In ...
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1answer
20 views

How does normalizing the response affect likelihood?

I have a vector of experiment outcomes, $Q$, and I assume that $Q_i$ are generated by a Gaussian distribution, i.i.d., such that the likelihood is the standard $$\mathcal{L}(q_1, ..., q_n) = ...
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0answers
11 views

MANOVA: What to do with a u-shaped DV?

I am hoping to get some advice on my analysis for my MSc dissertation. I will try to keep it short but please let me know if there is more information I can provide to make the situation clearer. ...
0
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1answer
30 views

How to efficiently simulate values from a multivariate normal given one of the components?

Suppose $X, Y_i$ for $i=1...n$ are standard normal variable but are also correlated so collectively they come from a multivariate normal distribution. Now the complication is what if I want to ...
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0answers
17 views

Common methods for transforming non-normal variable to close to normality

I have a list of time series which contain negative values. Right now I am transforming the time series to all positive values >0 and using the Box-Cox transformation to reduce non-normality. My ...
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0answers
26 views

Conditional expectation of error based on multivariate normal variables

I have the following situation; I know the "true" model behind my regression but I am intentionally omitting some variables/regressors to simplify the problem. Suppose the true model is: ...
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0answers
14 views

Confindence of test estimation for gaussian mixture model

A simple Gaussian Mixture approach: I have a learning data $ { (x_1,y_1),(x_2,y_2),...,(x_n,y_n) } $. For learning it, I use a Mixture of Gaussian model and after learning it, I estimate new data ...
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1answer
91 views

Test if two normally distributed random variables have the same mean

We have two independent random variables which follow normal distributions $X_1\sim \mathcal N(\mu_1,\sigma_1)$, $X_2\sim \mathcal N(\mu_2,\sigma_2)$. From the context, we have that $\mu_1\leq\mu_2$. ...
2
votes
2answers
93 views

Expected value of x in a normal distribution, GIVEN that it is below a certain value

Just wondering if it is possible to find the Expected value of x if it is normally distributed, given that is below a certain value (for example, below the mean value). Thanks,
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1answer
53 views

Normal distribution and independence

I was reading about white noise and it stated: Although $\varepsilon_t$ & $y_t$ are serially uncorrelated, they are not necessarily serially independent, because they are not necessarily ...
0
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1answer
65 views

How are percentiles distributed?

I was taking a look at this page, and I can't seem to understand why the frequency plot of the percentiles is uniformly distributed. Distances between percentiles are not equal, so why is the ...
2
votes
2answers
61 views

How can I compute $\int F(x \mid a,b)f(x \mid w,z) {}dx$ in closed form?

Suppose $F$ is the cumulative distribution function of the normal distribution with mean $a$ and standard deviation $b$, and suppose $f$ is the probability density function of the normal distribution ...
2
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0answers
26 views

Alternative to ANOVA (beginner)

I have run 15 experiments to compare the effect of different hormone combinations on the maturation on Xenopus oocytes (immature eggs). I am hoping to find the best performing variable. I have 4 ...
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1answer
23 views

What is the covariance when you know the covariance w.r.t. a common variable?

Say you know that ${\rm var}\Bigg( \begin{bmatrix} {\bf x}_1 \\ {\bf x}_2 \end{bmatrix}\Bigg) = {\bf \Sigma} = \begin{bmatrix} {\bf \Sigma}_{11} & {\bf \Sigma}_{12}\\ {\bf \Sigma}_{21} & ...
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0answers
30 views

The Dirty Dozen Triad Test and how it's scored

I'm trying to figure out some statistics to do with a test for a personality disorder called 'The Dark Triad'. The test is called the 'Dark Triad Dirty Dozen' (DTDD). The test is reported in ...
4
votes
1answer
52 views

Testing for normality in non-normal distributions with zero skewness and zero excess kurtosis

[This question was formerly called "On Non-normal distributions with zero skewness and zero excess kurtosis" and relabeled to better reflect its focus.] I am trying to write a little simulation using ...
0
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0answers
8 views

Analytic pdf for rotated 2D Gaussian with uncertain angle

I need to find an analytic form of the pdf of 2D Gaussian rotated with an uncertain angle, which itself is distributed according to 1D Gaussian. Let's say we have a zero-mean 2D Gaussian: $V \sim ...
2
votes
1answer
60 views

Why are coefficients estimates more accurate when using a hierarchical model?

I am reading a paper and I do not understand a bit: "We use a hierarchical model such that all slopes are calculated within the same model and the intercepts and slope components are normally ...
2
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0answers
16 views

Calculating ROC curve for two gaussian distributions with equal variance

I recall a simple formula in a paper which relates the distance between two gaussian distributions (what psychologists refer to as d') to the ROC curve under the assumption that both distributions ...