Questions tagged [normal-distribution]
The normal, or Gaussian, distribution has a density function that is a symmetrical bell-shaped curve. It is one of the most important distributions in statistics. Use the [normality] tag for asking about testing for normality.
6,009
questions
0
votes
0
answers
19
views
How is it possible to compare powers of normality tests?
The famous paper [1] compares the power of several normality
tests. Regardless of its result, my immediate question is, "How
is that even possible?"
Well, the definition of power in this ...
- 235
0
votes
0
answers
11
views
Complete Statistic for a family with finite r-th moment
Consider the family of all continuous distributions with finite $r-th$ moment (where $r \geq 1$ is a given integer). We denote this family as, $$\mathscr{P}_r=\{f:f \ \text{is a pdf and} \int|x|^rf(x)...
- 35
0
votes
0
answers
12
views
prior distribution for iid gaussian, with a known variance
I have been reading Pattern Recognition and Machine Learning by Bishop, and I have a question regarding the prior distribution of an iid Gaussian with known variance.
The relationship $\dfrac{n}{\...
- 8,647
1
vote
0
answers
14
views
Entropy of $\ell^p$ norm of multivariate Gaussian
If $X$ is n-dimensional standard Gaussian, is there an analytic expression for the differential entropy of the $\ell^p$ norm of $X$?
For the case $p=2$, the $\ell^2$ norm is exactly the chi ...
- 33
3
votes
0
answers
56
views
Best way to calculate max minus sum of random variables
I am trying to calculate the following probability distribution:
$Max(X_1, X_2,..., X_n) - \frac{1}{n} \sum _{i=1} ^{n} {X_i}$.
$X_i$ are iid random variables whose pmf is $\frac{C_{i-1}}{2^{2i-1}}$, ...
- 75
1
vote
2
answers
52
views
How to test for normality on paired samples between two treatments when the number of observations per treatment is unequal?
I have mice which need to poke a device to receive food across a one hour period. The device records the number of pokes and poke_time which is the duration that the mouse remained in the poke hole. ...
- 13
2
votes
0
answers
15
views
Identify the distribution of data [closed]
I did one experiment two times with the same subject. I want to find the repeatability of my result. repeatability in a sense is the fraction of behavioural variation that is due to differences ...
- 21
0
votes
0
answers
60
views
Normalizing Power Law Data [closed]
I am working with a dataset of ordinal data, denoted as $O$, to which I am applying a power law distribution (Zeta, Pareto, Yule-Simon, etc.) resulting in a dataset $P = \zeta(O)$ that appears roughly ...
- 101
0
votes
2
answers
40
views
How can we maintain asymptotic normality with slight change?
If $(X_n-\mu_n)/\sigma_n\rightarrow_{d} N(0,1)$ (i.e., $X_n$ is $AN(\mu_n,\sigma_n^2)$), I want to show the following two statements:
(1) $X_n$ is $AN(\bar{\mu}_n, \bar{\sigma}_n^2)$ if and only if $\...
- 1
0
votes
0
answers
28
views
normality testing [duplicate]
Good day,
I am a medical student currently working on a thesis focused on metabolic syndrome. My sample size is 45 individuals. I have encountered multiple challenges in assessing the normality of my ...
1
vote
1
answer
26
views
Calculating the mean and sd of a lognormal distribution from the log mean and log 5th percentile [duplicate]
I have a mean and 5th percentile value from river flow data that are assumed to follow a log-normal distribution. From these, I need to calculate the the mean and SD of the underlying normal ...
- 113
1
vote
0
answers
9
views
Stationary distribution of Markov chain with continuous state space [migrated]
I'm considering a Markov process with a continuous state space. Let $V(x)$ be a differentiable function, $\Delta t$ a fixed time step, and, at every step, set
$$x_{n+1}= x_n-\alpha \frac{dV}{dx}\Big|_{...
0
votes
0
answers
12
views
Comparing groups that are impacted to different extent by same truncation
Say I have results for a running event, where there is a cutoff time that is the same for all participants. I assume the results are normally or log-normally distributed, but truncated at the cutoff ...
0
votes
0
answers
22
views
Separating components of a likelihood maximization
Apologies for the naive question, but I have a problem I would like to solve.
Suppose I have a two dimensional likelihood of the form
$L \propto \exp\{-\frac{1}{2}\} \begin{bmatrix}x & y\end{...
- 1
0
votes
1
answer
46
views
Data transformation to normal distribution in analysis of variance (ANOVA)
Are there any methods to determine normality of data and then transform that data to normal distribution if it isn't normal?
The task has been set: It is necessary to conduct research via ANOVA and ...
- 3
0
votes
0
answers
14
views
VaR (value-at-risk) of a Normal Random Variable - Confused about Scaling in Derivation
In the book "Quantitative Risk Management: Concepts, Techniques and Tools - Revised Edition" (by McNeil, Frey, and Embrechts)", there is the following example (example 2.11):
Suppose ...
1
vote
1
answer
100
views
Poisson Regression and Linear Regression give the same error
I'll use the data azdrg112 from COUNT package. The los will be the response variable while ...
- 75
3
votes
1
answer
116
views
What is the probability of $P(X>Y>0)$ where $X$ and $Y$ standard normal distribution with correlation $\rho$?
What I have done is use Cholesky decompostion to get $X = \rho X^{1} + \sqrt{1 - \rho^2}X^{2}$ and $Y = X^{1}$ where $X^{1}$ and $X^{2}$ are independent standard normal. Then
$$
\begin{aligned}
P(...
- 31
3
votes
1
answer
219
views
Distribution of a sum of linear combinations of random variables, each drawn from a set of random variables
Question.
Let $X_1, X_2, ..., X_n$ be a set of normal random variables, each with variance ${\sigma }^2$ and mean 0. For each $i,j$ in pair in $X$, $Cov(X_i,X_j)=V$.
Further, let $Y_1, Y_2, ..., Y_m$ ...
1
vote
0
answers
24
views
Develop a model for theoretical best performances at different ultra running distances/times
Goal
Develop a model for theoretical best performance for running distances from marathon to around 1000 km. Partly to compare the strength of ultrarunning world records, but more importantly, to get ...
0
votes
0
answers
34
views
Fitting truncated sample to normal distribution with unknown mean & variance
I have data that is somehow truncated. It is a list of log best performances from events, where different events have different cutoff times.
How could it be possible to find the unknown mean and ...
0
votes
1
answer
39
views
How to derive the combined frequency distribution from two independent normally distributed population?
We have two different independent internal function calls for an application module, max of the time taken by two function calls individually determines how much the overall module takes. Individually,...
- 101
2
votes
0
answers
32
views
Gaussian linear regression with no noise
In short: is there an alternative expression for posterior on weights in a linear model, that works with no observation noise?
In Rasmussen & Williams "Gaussian Processes" they consider ...
0
votes
1
answer
62
views
Probability distribution of measurements and Parameters of measurements
I am new to statistics and recently learned about ISO guidelines for Accuracy & Precision and Uncertainty & Error. I have tried to plot a graph for what I have learnt including all the ...
1
vote
0
answers
54
views
Estimating a secret with before/after interchanging noises
$\newcommand{\Var}{\mathrm{Var}}\newcommand{\E}{\mathrm{E}}$
For $n+1$ iid "noise" variables $X_0,\dots,X_n$ from the normal distribution $\mathcal{N}(0,1)$ and a "secret" $s$ ...
- 11
3
votes
1
answer
53
views
Conditioning on two normal variables $E[x|y=y_0,x\leq k]$ and obtaining an analytical expression
suppose $x=a+e$ and $y=b+e$, where $e$ is normally distributed with mean 0 and variance $\sigma^2_e$. $a$ and $b$ are independent and normally distributed, both with mean $\mu$ and variance $\sigma^2$....
0
votes
0
answers
10
views
Is my explanation of how MLE works is correct? [duplicate]
The likelihood of observing the dataset we have (for some mean and variance) can be written as the product of the likelihood of observing each data point (since all the rows are independent). Now, ...
Siddhant Bashisth
3
votes
2
answers
539
views
Can I assume normal distribution?
I have calculated daily price returns of Bitcoin and plotted this data in the following way:
x-axis: count
y-axis: returns in %
I assumed the data had a normal distribution to calculate the
$$\rm ...
- 41
5
votes
2
answers
162
views
All uncorrelated marginals are independent: Only for joint Gaussian?
Let $X$ be a random vector in $\mathbb{R}^p$, where $p\geq 2$, with the following property: Any two uncorrelated marginals are independent. Formally:
(1) For any $\alpha,\beta\in \mathbb{R}^p$, if $...
- 213
0
votes
0
answers
7
views
Combining factors, represented as normal distributions, to one combined factor, normally distributed
I'm trying to combine the different factors that may affect running pace, such as GPS-measured distance, grade, terrain, heat and other factors (such as wind etc.). Each factor is represented as a ...
0
votes
2
answers
91
views
Will this converge to origin?
Suppose you have a diffusion of 100 points with the following iteration:
$$(x_{n+1},y_{n+1}) \sim \mathcal{N}\left((x_n,y_n), \frac{x_n^2 + y_n^2}{2} I_{2 \times2}\right)$$
This will make a high ...
- 111
0
votes
0
answers
45
views
Variance of powers of a standard normal random variable
To predict growth of money in a stock market I try to calculate expected return over a longer timeframe (e.g. 30 years) with a confidence interval. The simple math of taking an average stock market ...
- 139
1
vote
1
answer
37
views
The Tsallis entropy of generalized Gaussian distribution
I would like to discuss the computation of the Tsallis entropy for the generalized Gaussian distribution. From the paper in the link https://www.sciencedirect.com/science/article/pii/S0167947322000822....
- 85
1
vote
0
answers
42
views
What is the variance on the ratio of dependent random variables
I have the data on thousands of emission lines such as the one shown in the figure below. A single emission line covers $N$ pixels (11 in this example). Because the data come from counting photons, $...
- 11
0
votes
0
answers
21
views
How to calculate the likelihood for a normal distribution N(theta, 1) if we only know the maximum of a sample?
Assuming iid samples x ~ N(theta, 1), we have a sample of 5 observations with maximum value = 3. How to calculate the likelihood?
- 1
1
vote
0
answers
19
views
Unifying multiple sample data (meta-analysis)
I would like to unify multiple sample data (of different sizes) into some "unified sample" to evaluate its collective variance. Is this something statisticians do? I thought of unifying ...
- 11
1
vote
1
answer
48
views
Predictive Distribution in Gaussian Process for Machine Learning
I am reading Gaussian Process for Machine Learning equation 2.9, where it is deriving the predictive distribution
$$p(f_* | \mathbf{x}_*, X, \mathbf{y}) = \int p(f_* | \mathbf{x}_*, \mathbf{w}) p(\...
- 11
0
votes
0
answers
32
views
In a skewed sample with a large n, does Central Limit Theorem dictate that a t-test can be used, even if the mean cannot be interpreted? [duplicate]
I understand that, in the case of a highly skewed population and sample, the sampling distribution of the mean can still be normally distributed if the sample size is large, according to Central Limit ...
- 95
3
votes
2
answers
256
views
Why is it the convention to take equal tails in a two-tailed test with a statistic following a symmetric distribution?
Is there a particular reason for conventionally dividing the tails equally in a two-tailed test? Consider an $\alpha$ level test with a statistic following standard normal distribution. Then, why do ...
- 187
0
votes
0
answers
27
views
Distribution looks roughly normal on a q-q plot, but has a p-value of 0.0 for the Shapiro-Wilk normality test. How to interpret? [duplicate]
The distribution is as follows:
However the Shapiro-Wilk test yields a p-value of 0.0 and a W statistic of 0.9. There are over 7,000 values in the sample.
Note, the quantile values have been ...
- 111
4
votes
2
answers
135
views
Expectation of a normal variable given that a signal is below a certain value
If $\tilde{u}$ is a normally distributed random variable with mean $q$ and variance $\sigma$, and $s=\tilde{u}+\tilde{e}$ where $e$ is also normally distributed with mean 0 and variance $\sigma_s$. $\...
0
votes
0
answers
13
views
How to determine significance for a Corrado rank test in an event study?
I apologise if this is a stupid question (it feels stupid tbh). I am currently doing an event study and my abnormal returns are not normally distributed. I am now in the process of performing a ...
0
votes
0
answers
31
views
Understanding usage of quantile function (norm.ppf), passing p vs 1-p
I was given a question related to the quantile function using scipy.stats.norm.ppf, along with its solution - which I don't understand.
I'm changing the question ...
- 353
0
votes
0
answers
38
views
Why does the reparameterization trick work when some components are still stochastic? [duplicate]
I am trying to understand the reparameterization trick. I got some intuition while looking at this popular question, but I still feel largely confused. I am putting my understanding and doubts here ...
- 500
1
vote
2
answers
51
views
Why do Poisson regression and Linear regression give the same error? [closed]
I use the badhealth data as an example, I'm modelling the number of visit (doctor) based on the health condition and the age as two features. Mean absolute error is ...
- 75
1
vote
1
answer
41
views
Finding the Variance of the MLE Variance of a Joint Normal Distribution
I have a random sampling of $Z_1,...Z_n$ from a normal distribution $N(\mu,\sigma^{2})$. I am considering them within a joint likelihood function.
I know that the MLE ($\hat\sigma^{2}$) of $\sigma^{2}$...
- 13
1
vote
1
answer
43
views
Deriving covariance of joint distributions of MVN [Linear Gaussian systems?]
Let $z$ ∈ R^L be an unknown vector of values, and $y$ ∈ R^D be some noisy measurement of z. We assume these variables are related by the following joint distribution
$p(z) \sim N(z|\mu_{z}, \Sigma_{z})...
- 111
5
votes
1
answer
237
views
What is the substantive meaning of one statistical test is more powerful than another?
There are some research claims that one statistical test is more powerful than another. For example, a highly cited study states:
Results show that the Shapiro-Wilk test is the most powerful ...
- 408
1
vote
0
answers
14
views
Interpretation of Anderson–Darling test
Assume that I am using Anderson–Darling test to evaluate whether a given sample of data is drawn from a normal distribution with some mean and standard deviation.
If you accept the null hypothesis in ...
- 248
0
votes
0
answers
20
views
Computing an integral that reduces to $\mathbb{P}[X>Y]$
Problem
Evaluate $$I=\int_{-\infty}^\infty \frac{e^{-\frac{1}{2}\left(\frac{x-\mu)}{\sigma} \right)^2}}{\sigma \sqrt{2 \pi}}\frac{1}{1+e^{-x}}\, \mathrm{d}x$$
My attempt
Now the first part of the ...
- 285