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Questions tagged [t-distribution]

t is the distribution of the t-statistic that results from a t-test. Use this tag only for questions about the distribution; use [t-test] for questions about the test.

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name for distribution of sample mean before standardization to t-distribution [closed]

I’m re-learning a very basic statistics of standard error of mean. When population variance is known, the distribution of sample mean is normal distribution, according to the central limit theorem. On ...
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Why confidence interval for proportion uses z instead of t score? [duplicate]

The confidence interval (Wald interval) for the parameter $p$ of a binomial distribution is computed from the approximation by a Normal distribution: $$ \ p ~~ \approx ~~ \hat p \pm \frac{\; z_\alpha\ ...
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Conjugate Prior for Student T distribution with known degrees of freedom

Somebody asked a question about a conjugate prior distribution for Student-t distribution with unknown degrees of freedom. It was answered that there are no conjugate prior distribution in that case. ...
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Thought and doubt about Student's t-distribution and confidence interval

Let's say I have $N$ observations $X_1,X_2,\ldots,X_N$, where $X_i\sim\mathcal\mu,\sigma^2$, $\forall\,i\in\mathbb{N}$, where $\mu$ and $\sigma^2$ are unknown. I want to predict $X_{N+1}$ and ...
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4 votes
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Simulating from fitted t-distribution mgcv

My question is related to the following post: Extracting the degrees of freedom of t distribution of a GAM I have a dataset of y-values that I am fitting using the gam-function in mgcv. The ...
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Expectation of Mahalanobis Distance and its logarithm

Suppose: \begin{equation} X \sim \mathcal{N}(X, \mu, \Sigma_x) \text{ st. } \Sigma_x \sim \mathcal{IW}(\Sigma_x; \Psi, v) \end{equation} Where $\mathcal{IW}$ is the Inverse-Wishart distribution. This ...
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T Distribution and CLT

By definition, the T distribution is the ratio of standard normal variable and sqrt of scaled $\chi^2$ variable. The "popularized" version of (one sample) t statistic goes like this: $\frac{\...
Kaiwen Wang's user avatar
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1 answer
101 views

Relationship Between Chi-Square/Gamma & t/lst distributions?

I'm trying to understand $\chi^2_n$ & $\Gamma(\theta, k)$ distributions. Currently I believe they're comparable to t (aka $t_v$) & location-scale-t (aka $lst(\mu, \sigma^2, v)$) distributions ...
profPlum's user avatar
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2 votes
1 answer
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How can I calculate the minimum required sample size for a one-tailed paired t-test, testing whether the difference is greater than 5?

This Cross Validated answer states that it's valid to do a paired t-test to test whether the difference is greater than a certain value (e.g. 5) by changing the specified value of 𝜇 in the null/...
Alice's user avatar
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1 answer
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A/B Test from two distributions with unknown means

I will run a experiment to check if a given change, represent some significance in user interactions. I have two sets, sampled from two groups, one called control ...
Lin's user avatar
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Product of Two t-distribution Formulas

Does the product of two t-distribution formulas with same degrees of freedom simplify? $T_v(x; \mu_1, \Sigma_1)T_v(x; \mu_2, \Sigma_2) =\ ?...$ In the normal case it simplifies to: $\mathcal{N}(x; \...
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Mean of normal follows a T distribution

Suppose: $x \sim \mathcal{N}(x; \mu, \Sigma) \;\;\;$ st. $\;\;\; \mu \sim T_{v}(\mu; k, M)$ Where $T$ is the $t$-distribution with v degrees of freedom, location $k$, and shape $M$. Then, is there a ...
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Resulting univariate marginal distributions are not $t$ distributed - why? (S Wood Core Statistics)

In the Core Statistics by Simon Wood it says: "If we replace the random variables $Z_i\sim_\text{i.i.d.} N(0,1)$ with random variables $T_i \sim_\text{i.i.d.} t_k$ in the definition of a ...
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Understanding of Gamma distribution as precision prior in Bayesian inference for Gaussian

Christopher M. Bishop in his book "Pattern Recognition and Machine Learning" nicely explains where does Student t-distribution $St(x|\mu,\lambda,\upsilon)$ originate into. In Chapter 2, it ...
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posterior predictive of a normal distribution with normal prior over mean and Gamma prior over precision

What is the posterior predictive of a normal distribution with normal prior over mean and Gamma prior over precision. Thus, what is the distribution of x given: \begin{equation} x \sim \mathcal{N}(x; \...
Snowy Baboon's user avatar
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Central Limit Theorem for t(2)-distributed random variables

Let $X_k \overset{iid}{\sim} t(2)$. Is there any limit theorem about $\bar{X}$? I know $\text{Var}(\bar{X})$ doesn't exist, so I cannot use classical CLT. But I believe there must be other means to ...
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Finding probability that the mean of a sample is below a certain limit

I have a complicated physical model that can produce a certain quantity (real-valued) for a large (in the 1000's) number of points (N). We assume that mean and variance of the model output is a good ...
FrenchKheldar's user avatar
1 vote
1 answer
48 views

If two t-statistics have the same degrees of freedom, is the p-value of the larger t-statistic greater than the p-value of the smaller t-statistic?

If I'm understanding correctly, calculating p-values from a t-statistic is just an integral of the t-distribution pdf, and so if two t-stats have the same number of degrees of freedom then the larger ...
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In simple linear regression model, why do we calculate the confidence interval for slope parameter using t-distribution? [duplicate]

I'm taking a regression analysis course and we were studying simple linear regression. I've understood how slope $$ \hat\beta_1 follows \space N(0,\sigma^2 / S_{xx}) $$ and is normally disributed. And ...
aroma's user avatar
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Why does the t-SNE paper claim that "large clusters of points that are far apart interact in just the same way as individual points"?

In the original t-SNE paper, the authors explain the use of the t-distribution with one degree of freedom (i.e. Cauchy distribution) for the map points, $(1 + |y_i - y_j|^2)^{-1}$, as follows: ...[it]...
Denziloe's user avatar
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14 votes
1 answer
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Intuitive explanation for the fat tails of the t-distribution

Given some standard assumptions, the test statistic $$ \frac{\Delta\bar{X}}{\sigma/\sqrt{N}} $$ is normally distributed if $\sigma$ is known and t-distributed if $\sigma$ has to be estimated from the ...
monade's user avatar
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Sample covariance of t distribution and degree of freedom

If $X$ is a P by N size matrix, $X_{ij} \sim N(0,\sigma_i^2)$ if I standardize this X matrix with sample mean and sample variance (assuming I don't have access to the population mean and variance) I ...
maddy's user avatar
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206 views

Can we transform a t-distribution into a normal distribution? [closed]

We know that any normal distribution can be transformed into a t-distribution as shown in this post: Transformation of any normal distribution into a standardized t-distribution My question is can we ...
Hepdrey's user avatar
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Bayesian inference on model parameters from summary statistics alone

Consider quantities $y_1,y_2,\dots,y_p$, for the $j$th of which we have $n_j$ measurements $y_{1j},y_{2j},\dots,y_{n_jj}$. Unfortunately, I do not have access to the raw data $y_{ij}$ -- only to the ...
SoupyTwist's user avatar
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Minimizing the NLL of a t-distribution derived from a NIG prior

My question concerns this paper which is a little too succinct for me to understand. The context is the following. Suppose $y$ is Normal distributed, with a Normal-Inverse-Gamma prior, $$ y \sim N(\mu,...
stevew's user avatar
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The distribution of the square of a Student's t-distributed random variable

The following may not be very difficult, but I'm struggling to get started. $T_k = \frac{Z_0}{\sqrt{\frac{1}{k}(Z_1^2 + \dots + Z_k^2)}}$, where $Z_0, \dots$ are i.i.d. standard normal random ...
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Calculating confidence intervals for coefficients in ordinal logistic regression

I've fitted an ordinal logistic regression model in R using MASS::polr and I am looking to compute confidence intervals for the fitted coefficients. Here is the (...
damtheduck's user avatar
2 votes
1 answer
104 views

An approximate confidence interval for the $\alpha$ parameter of a Pareto Type II distribution when $\lambda$ is known

The Pareto Type II distribution, also known as the Lomax distribution, has the following density, $$f(x|\alpha,\lambda)=\frac{\alpha\lambda^{\alpha}}{(\lambda+x)^{\alpha+1}}, \qquad x>0,\ \alpha>...
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How many sample do we need for normality of t-ratio? [duplicate]

I am currently learning about confidence intervals for the population mean. Assume we do not know the variance of the population. Let $\bar{x}$ be the sample mean, $s$ be the sample variance and $n$ ...
pele's user avatar
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Any meaningful interpretation t-distribution, when rescaled using the sample SD?

To visualize p-values or confidence intervals, the t-distribution is sometimes rescaled using the sample standard deviation and then centered at a certain value. To be more specific, consider drawing ...
arb's user avatar
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9 votes
1 answer
630 views

Kullback–Leibler divergence between two multivariate t distributions with different degrees of freedom?

I want to calculate the Kullback–Leibler divergence between two multivariate $t$ distributions with different degrees of freedom (say $\nu_1$ and $\nu_2$), but same location and scale matrix, for ...
Student's user avatar
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1 vote
2 answers
71 views

Interpretation of distribution that appears when calculating CI for population mean

Let $X \sim \mathcal{N}(\mu, \sigma)$ be the model for a normally distributed population, described by the probability density function $f_{X}(x; \mu, \sigma)$. We can denote $\mathbf{X} = (X_1, X_2, \...
ivan's user avatar
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1 answer
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Population means difference using t-distribution

I wanted to understand the solution to a question I got incorrect. In particular given two independently sampled iid data of different sizes (m and n), possibly different means ($\mu_1, \mu_2$), and ...
JustBlaze's user avatar
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23 views

How to estimate regression coefficients if the errors are t-distributed?

I estimated the linear regression using ols but the errors turned out to be t-distributed with df=3 according to q-q plot, I already know that gauss-markov theorem still assume the coefficients to be ...
Yarh's user avatar
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1 vote
3 answers
777 views

Should the formula for test statistic have an absolute value?

I have been given a formula for calculating a t test statistic as $$t_{n-1} = \frac{ \bar{x}- \mu_0}{\frac{S}{\sqrt{n}}} $$ Where $\bar{x}$ is the sample mean, $\mu_0$ is the hypothetical true mean, $...
Kirsten's user avatar
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3 votes
1 answer
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Fraction-manipulation between a Gamma and Student-t

I'm working on a course problem, Suppose that $\textbf{x}=\{x_1,\dots,x_n\}$ and $\textbf{y}=\{y_1\dots,y_m\}$ are independent random samples from $\text{N}(x|\mu_x,1/\lambda)$ and $\text{N}(y|\mu_y,...
mjc's user avatar
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2 votes
0 answers
88 views

Trying to understand Fisher's geometric derivation of t distribution

I'm trying to understand this answer: https://stats.stackexchange.com/a/151969/25186 which details Fisher's geometric derivation of the t-distribution. There are some loose ends in my understand I'm ...
ryu576's user avatar
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5 votes
3 answers
624 views

Should the t-statistic (not data) be normally distributed for using the t-test?

Reading the introduction of English Wikipedia's article about the t-test, I was confused by this statement: It is most commonly applied when the test statistic would follow a normal distribution To ...
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Standard deviation of two assets under normal distribution and t-distribution

I have the following two questions it might consider easy but I need to understand the intuition behind it Assume I have two financial assets , my portfolio return model $$ \begin{align} \label{s2} w^...
A.F.R.S2022's user avatar
1 vote
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99 views

Use of the Student T distribution for the t-statistic

The variable $T$ is given by the classic form of the one-sample t-test the population mean under a hypothesis such as $H_0: \mu = \mu_0$ $$T = \frac{\bar{X_n}-\mu_0}{S/\sqrt{n}}$$ where $\bar{X_n} = \...
Mr Saltine's user avatar
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1 answer
59 views

What is the best method to test the null hypothesis on volatile data?

I am trying to find a good method to test the null hypothesis(H0) on two unpaired samples. Those samples come from two different HTTP Servers and the unit I'm using is req/30s (requests concluded in ...
Rafael's user avatar
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1 vote
0 answers
69 views

Transform multivariate Gaussian into to multivariate t-distribution

A multivariate t-distribution random variable $X \sim St(\mu_X, C_X, \nu_X) \in \mathbb{R}^n$ can be constructed as follows: \begin{equation} X = \sqrt{\frac{u}{\nu_X}} Y + \mu_X \end{equation} where ...
Mathieu le provost's user avatar
1 vote
0 answers
86 views

Moments of a Ratio distribution (Normal variable / Non-central chi variable)

Let $[x_1, x_2, x_3]$ are three independent points in Cartesian space which are Gaussian distributed with a non-zero mean and identity covariance. I need to calculate the following expectations, \...
Nikhil Sharma's user avatar
1 vote
1 answer
59 views

Meaning of capital $S$ in Student's "The Probable Error of a Mean"

I'm reading through Student's "The Probable Error of a Mean" from 1908 and I'm not sure what notation he is using. He says that if $s$ is the standard deviation from a sample $x_1 x_2 \ldots ...
Vityou's user avatar
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2 votes
2 answers
545 views

Using normal distribution to approximate t distribution in importance sampling

The question is Exercises 6 and 7 regarding importance sampling on page 273 of Bayesian Data Analysis 3 http://www.stat.columbia.edu/~gelman/book/BDA3.pdf. Exercise 6 approximate a normal distribution ...
Statisfun's user avatar
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1 vote
1 answer
471 views

How to prove $(\hat{X}-\mu)/(\hat{S}/\sqrt{n})$ is student t with $n-1$ degrees of freedom if $X_i$ are iid $N(\mu, \sigma)$?

It is commonly stated that if $X_i$ are iid $N(\mu, \sigma)$, then with $\hat{X}$ the sample mean, and $\hat{S}$ the sample error (sample standard deviation), then $\frac{ \hat{X}-\mu}{\hat{S}/\sqrt{n}...
travelingbones's user avatar
3 votes
1 answer
80 views

How to eliminate constant to derive the decision rule in terms of the sufficient statistic $\bar{X}$ for normal distribution means hypothesis test?

Suppose that we have a random sample, of size $n$, from a population that is normally-distributed. Both the mean, $\mu$, and the standard deviation, $\sigma$, of the population are unknown. We want to ...
user avatar
1 vote
0 answers
333 views

How to compute empirical confidence intervals correctly?

Given a (moderate) number of values which can be assumed to be normal distributed but with unknown mean and std-dev, I want to compute a confidence interval (say 99% confidence). My naive (not a stats ...
Kat Branchman's user avatar
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0 answers
23 views

Can the multivariate-t distribution have density greater than 1? [duplicate]

I'm working with a Python implementation of the Multivariate T distribution, and I've noticed when I evaluate the PDF at certain points, the likelihood returned is > 1. This is causing issues in ...
Addison's user avatar
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3 votes
1 answer
482 views

Why does t-statistic for arbitrary parameter follow t distribution?

I understand that given a standard normal variable $Z$ and a $\chi^2$ random variable $V$ with $\upsilon$ degrees of freedom that \begin{align*} T := \frac{Z}{\sqrt{V/\upsilon}} \end{align*} ...
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