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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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; \...
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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 ...
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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
...
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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]...
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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 ...
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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 ...
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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 ...
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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 ...
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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,
$$
y \sim N(\mu,...
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The distribution of the square of a Student's t-distributed random variable
The following may not be very difficult, but I'm struggling to get started.
$T_k = \frac{Z_0}{\sqrt{\frac{1}{k}(Z_1^2 + \dots + Z_k^2)}}$, where $Z_0, \dots$ are i.i.d. standard normal random ...
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Calculating confidence intervals for coefficients in ordinal logistic regression
I've fitted an ordinal logistic regression model in R using MASS::polr and I am looking to compute confidence intervals for the fitted coefficients.
Here is the (...
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An approximate confidence interval for the $\alpha$ parameter of a Pareto Type II distribution when $\lambda$ is known
The Pareto Type II distribution, also known as the Lomax distribution, has the following density,
$$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 $n$ ...
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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 ...
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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 ...
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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 $\mathbf{X} = (X_1, X_2, \...
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Population means difference using t-distribution
I wanted to understand the solution to a question I got incorrect.
In particular given two independently sampled iid data of different sizes (m and n), possibly different means ($\mu_1, \mu_2$), and ...
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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 ...
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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, $...
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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,...
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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 ...
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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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(Correlation like) effect size for relationship between two Student t-distributed variables
I have a data sample with two variables that are seemingly t-distributed. I would like to calculate something like a correlation coefficient to express the effect size for the relationship between the ...
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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
$$
\begin{align} \label{s2}
w^...
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Use of the Student T distribution for the t-statistic
The variable $T$ is given by the classic form of the one-sample t-test the population mean under a hypothesis such as $H_0: \mu = \mu_0$
$$T = \frac{\bar{X_n}-\mu_0}{S/\sqrt{n}}$$
where $\bar{X_n} = \...
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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 ...
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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 ...
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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,
\...
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1
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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 ...
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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 ...
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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}...
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Statistician Gosset and his work trying to prove the existence of God [closed]
I have read that the inventor or discoverer of the student t-distribution, a person whose surname is Gosset, was trying to prove the existence of God when he made his discovery.
Is this just something ...
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answer
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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 ...
user318514
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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 ...
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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 ...
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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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votes
1
answer
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Why do Normal distribution and T distribution have different standard errors?
In the context of t-distribution confidence intervals, when computing the standard error of the difference of means when population sizes are uneven, the harmonic mean of the sample sizes must be used....
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How to calculate conditional probability on student multivariate distribution
I have a multivariate student distribution fitted on some data on 4 dimensions (so I know the parameters).
I am trying to calculate the $P(X_4\le x_4| X_1=x_1, X_2=x_2, X_3=x_3)$ but falling short.
I ...
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Proof of the Student t-test for independent samples drawn from the same normal distribution when $\mu \neq 0$
I'm following the proof in Cramer's book Mathematical Methods of Statistics, $\S 29.4$. There it is assumed that we have two independent samples $x_1,\ldots, x_{n_1}$ and $y_1,\ldots,y_{n_2}$ drawn ...
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T-distribution problem
I am working through the exercises of Martin Sternstein's Statistics (Barron's College Review Series) ... struggling with Exercise 4 in Chapter 14.
Given are the weight lifting strengths of 9 people ...
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Product of two independent Student distributions
What is the product of two independent student t distributions?
In which case does this product product result in another t distribution?
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answer
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T-distribution parameters with QRM package
I am fitting a t-distribution on some data I have using the fit.st function from the QRM package.
The function returns 2 set of ...
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When does the sum of two $t$-distributed random variables follow a $t$ distribution?
In the scope of a project, I need to find the sum of two independent $t$-distributions.
I know that in the general case, the sum of two $t$-distributed random variables is not $t$-distributed. However,...
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Deriving the CDF of student $t$-distribution
I am trying to derive the cumulative density function(cdf) of $t$-distribution, define as in
https://en.wikipedia.org/wiki/Student%27s_t-distribution#Cumulative_distribution_function
I derive it by ...
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answers
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Understanding the relationship between a 'sampling distribution of a statistic' and a 'population distribution' in the context of a 'hypothesis test'
I would like to confirm that I am understanding the relationship between a sampling distribution of a statistic (an example of a 'statistic' would be a sample mean $\bar{x}$) and a population ...
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T-Statistics vs. Z-Statistics [duplicate]
If Z-statistics, when population variance is unknown, is a close approssimation of the T-statistics, which is better suited for the case, why do we overcomplicate things by using Z if sample size is ...
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Variance estimate for Student's t-distribution with heavy tails
The variance of a Student's t random variable when the degrees of freedom $\nu$ is greater than $2$ is $\nu/(\nu-2)$. In R, when I try to estimate the variance using the usual estimator $$1/(n-1)\sum_{...
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