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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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What does it mean exactly to divide a distribution by another distribution?

In the notes I'm working through, distributions are often "divided" by other distribution, and while I sort of understand what is meant, i would rather a rigorous explanation. Let me provide an ...
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Inconsitency in comparing log-likelihoods of z- and t-values with appropriate distributions?

I am working through "Glen McPherson: Applying and interpreting statistics", where he introduces the t-distribution on p.125. As a small experiment i tried to do the following (pyhton, scipy, numpy, ...
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MQL5: Question regarding MathProbabilityDensityT()

I use the MathProbabilityDensityT() function to get the p-value from the t-statistic and the degrees of freedom such as: ...
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How does software calculate exact p-values [duplicate]

I have calculated my t-value = 2.588 and the number of degrees of freedom is df=8. Looking up the t-value in a t-distribution ...
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1answer
33 views

How to calculate probability using T distribution in R and When to use T distribution

Can someone help me to solve below question using R programming. A Government company claims that an average light bulb lasts 270 days. A researcher randomly selects 18 bulbs for testing. The ...
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2answers
28 views

Can we replace the t-Student distribution by the Normal distribution in this context?

As far as I have studied, given a normal random sample, we can build the confidence interval of the mean $\mu$ if we know the variance through the relation \begin{align*} \frac{\sqrt{n}(\overline{X}-\...
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Adjusted Pearson Goodness-of-Fit Test - Rugarch Package

I fitted a GARCH(1,1), GARCH-M and EGARCH of first order (using maximum likelihood) to my return dataset using both, Gaussian normal and Student-t distribution assumption for the error term. When ...
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24 views

Is this a t-distribution problem?

The problem says that the probability of student failure is 0.4 A random sample of size 10 is taken from this population. I'm asked to find the probability that, at most, 30 % of students failed. I ...
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Conjugate Prior for Student T distribution

Does the Student T distribution have a conjugate prior distribution? If so, what is it and what are the parameters?
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63 views

How is Student's t-distribution related with this similarity/probability equation between data points?

In the t-SNE paper "Visualizing Data using t-SNE" and a Deep Embedded Clustering (DEC) approach "Unsupervised Deep Embedding for Clustering Analysis", they both use the Student t-distribution to ...
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How to relate beta CDF to student-t CDF? [duplicate]

We can relate the student-t and beta distributions as such: If $X$ has a Student's t-distribution with degree of freedom $\nu$ then one can obtain a Beta distribution: $$\frac{\nu}{\nu + X^2} \...
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How to approximate the student-t CDF at a point without the hypergeometric function?

Is there a way to closely approximate the CDF of a student-t distribution at a point $x$ without involving the hypergeometric function? For example, by using a series expansion, or expressing the CDF ...
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What's the relationship between degrees of freedom of t distribution and tail exponent (alpha) of Pareto distribution?

I'm going to generate a set of data from a T distribution and truncate the body(so that we make it approximately Pareto distributed) of it and estimate the tail exponent(shape parameter) of the ...
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Distribution of multivariate “$Z$-score”?

Suppose $\mathbf{X}_1, \dots, \mathbf{X}_n \sim N_p(\mathbf{\mu}, \Sigma)$ where $\mu \in \mathbb{R}^p$ and $\Sigma$ is a $p \times p$ covariance matrix. Suppose $\hat{\Sigma}$ is the sample ...
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Linear regression with error dispersion dependent on the independent variable

Suppose $y=ax+z$ where $x, y, z$ are random variables with range in $\mathbf R$, $\mathbf E[x]=0$, the probability distribution $p(z|x)$ is 1) normal distribution $N(0,\sigma(x)^2)$ with mean $0$ ...
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Which correction do I need to get a measurement uncertainty from the sample standard deviation?

I'm an experimental physicist who mainly needs statistics for the calculation of measurement uncertainties and confidence intervals. Since my results are usually normally distributed, I simply take $N$...
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What are the values of a student's distribution (k, µ and σ) when x1 = 10, x2 = 10 and x3 = 10? [duplicate]

I have a sample that contains only 3 values that are x1 = 10 x2 = 10 x3 = 13 Remark: I know that x3 value is not what is written in title I will calculate the ...
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How to “honestly” calculate likelihood of 2D normal with small sample size? [closed]

I estimate covariances from data and want to calculate likelihood. For 1D case I know - if the sample size is $<40$, I use Student's t-distribution to calculate likelihood of the data since my ...
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How should I calculate mean CI - from raw data or mean values?

I have experimenatal data from medical industry. In the experiment 4 observations were weighted, each one 2 times for more precise measurement. As a final weight, a mean from each pair is reported. ...
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2answers
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Existence of $E(X^2)$ when $X$ has the pdf $f(x)= \frac{1}{(2+x^2)^{3/2}}$

In a competitive exam, I came across an objective question which says Let $X$ be a continuous random variable with the probability density function $$f(x)= \frac{1}{(2+x^2)^{3/2}}\quad,\,-\infty&...
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Gradient signifiance test with tolerance and measurement uncertainties

I'm trying to test whether a likelihood has converged to walk around a single number or if it is increasing. In my (EM) algorithm, the noiseless likelihood cannot decrease, only ever increase. In my ...
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One sided test $H_0:\mu=0$ or $H_0:\mu\leq 0$? [duplicate]

I want to test $$H_0: \mu \leq 0 \,\,\,\,\,\, vs \,\,\,\,\,\, H_1: \mu > 0.$$ I am using a t test, so the statistic $T$ has $\nu$ degrees of freedom which depend on the sample size. What is ...
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Understanding this expression of the multivariate t-distribution

I found this SO post which expresses the PDF of a multivariate t-distribution in terms of the gamma and normal distribution in python as follows $$ G = \Gamma (k = \nu /2 ; \theta = 2 / \nu)\\ Z = N (...
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Distribution of coefficient on the error correction term in ECM and VECM

According to statistic academic literature, the cointegration test on coefficient $\alpha$ of the error term included in ECM or VECM does not follow a standard distribution. My question is: If so, ...
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Degrees of freedom when manually doing (Welch) T-test?

I have two samples A and B and want to test if the (Pearson) auto-correlation of A is greater than that of B. So far I've computed the two autocorrelations, $r_a$ and $r_b$ and found their standard ...
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1answer
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Confidence intervals over mean difference with unknown but equal variance

The problem is the following: I have 2 Stores, called Store1 and Store2 I take a sample of the number of item sold in each store over a certain period of time From the first store the sample has ...
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1answer
64 views

Where is t-distribution used in t-SNE?

I am trying to learn the dimensionality reduction using t-SNE technique. After some videos and explanation I understood the idea behind it. But I am not getting where the t-distribution is used behind ...
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Do we assume a t distribution for the estimate of the difference of normal distributions?

I am studying hypothesis testing. When performing a one sample t test, we assume a t distribution for the sample mean estimates of the true mean. When conducting a two independent sample test, it ...
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1answer
36 views

95% limits of a normally distributed parameter

How do I find the 95% limits of the population distribution of a normally distributed parameter? I've taken the mean and SD from 10 different readings of the parameter. Will the 95% limits be mean +/-...
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1answer
113 views

Find the pdf of $\frac{Z_1}{\sqrt{(Z_1^2+Z_2^2)/2}}$ where $Z_1,Z_2$ are i.i.d standard normal

Given $Z_1, Z_2$ are i.i.d standard normal random variables. Let $$V:=\frac{Z_1}{\sqrt{(Z_1^2+Z_2^2)/2}}$$ Derive the pdf of $V$. The numerator and denominator of $V$ are dependent, so the square ...
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What is the relationship of student t-distribution and gaussian function [duplicate]

The shape of student t-distribution looks pretty much like a Gaussian function, what is their relationship? Can t-distribution be represented by a closed form formula?
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multivariate Student's t distribution: intuition for non-independence?

Consider a multivariate Student's t distribution, with parameters $\nu$ (d.f.), $\mu$ (location) and $\Sigma$ (shape). Does anyone have a good intuition for the individual components not being ...
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What kind of distribution is this?

I'm sorry is this is too obvious, but I'm having a hard time trying to find a distribution for my data. It is clearly not a normal distribution. It does not seem to be skewed, but seems to have fat ...
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1answer
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Converting a prior probability of the null into t-test parameters

Suppose I am in the rare situation where I know for certain the prior probability of the null hypothesis. And yes, I mean the marginal probability of the null (not the probability of the null, ...
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2answers
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How do the t-distribution and standard normal distribution differ, and why is t-distribution used more?

For statistical inference (e.g., hypothesis testing or computing confidence intervals), why do we use the t-distribution instead of the standard normal distribution? My class started with the standard ...
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Derive p value from t distribution [duplicate]

I have difficulty of understanding p-value from An Introduction to Statistical Learning by Gareth James • Daniela Witten • Trevor Hastie Robert Tibshirani (2015) : 67 Consider simple linear ...
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Intuitive explanation for t vs z in confidence intervals

Skip to the conclusion if TL;DR What is an intuitive explanation for what a t-score gives you when computing confidence intervals? Background of my understanding of confidence intervals I just ...
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1answer
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Simulation of t copula in Python [on hold]

I am trying to simulate a t-copula using Python, but my code yields strange results (is not well-behaving): I followed the approach suggested by Demarta & McNeil (2004) in "The t Copula and ...
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1answer
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Find $\lim_{n \downarrow 1} t_{n-1, \alpha/2} / \sqrt{n}$ and prove the limit

This question is related to another recent question on CV.SE relating to the weighting function that commonly appears in confidence intervals for a population mean:. $$w(\alpha, n) \equiv \frac{t_{n-...
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1answer
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How is the family of distributions with PDF proportional to $(1+ax^2)^{-1/a}$ called?

Consider a family of distributions with PDF (up to a proportionality constant) given by $$p(x)\sim \frac{1}{(1+\alpha x^2)^{1/\alpha}}.$$ How is it called? If it does not have a name, how would you ...
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Prove that $t_{n-1, \alpha/2} / \sqrt{n} \rightarrow 0$ as $n \rightarrow \infty$

Statistical problems involving confidence intervals for a population mean can be framed in terms of the following weighting function: $$w(\alpha, n) \equiv \frac{t_{n-1,\alpha/2}}{\sqrt{n}} \quad \...
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2answers
393 views

Confidence interval for the mean - Normal distribution or Student's t-distribution?

I have taken $N$ measurements $x_i$ of the time taken by a computer program to run a particular calculation, and would like to calculate a 95% confidence interval for the mean runtime. We can ...
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37 views

Studentization ad pivotal quantity for the slope in simple linear regression

Take the usual linear model: $$ Y=\alpha+\beta X + e$$ In good hypothesis (IID sample, normal error assumption $e \sim N(0,\sigma_e^2) $ and homoschedasticity) the distribution of the sample slope $\...
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Can BIC prescribe a different model specification after changing unconditional distribution assumption?

I modelled an ARMA GARCH via minimization of BIC. The result was in accordance with the simple ARMA model i.e. both prescribed AR(1) as the optimal (in BIC terms) conditional mean model. By looking at ...
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MLE for Linear Regression, student-t distributed error

Say I have a $n$ pairs $(X_i, Y_i)$ and I have the relationship $$Y_i = \beta_0 + \beta_1X_i +\epsilon_i$$ where the error terms $\epsilon_i$ are iid from a student t-distribution with constant (known)...
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meta-analysis of linear regression coefficients

I have the summary results of a linear regression model with 26 covariates which has been fitted to two independent samples (with different sizes). Now I am planning to use the linear regression beta ...
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Student t - critical value [duplicate]

i need critical value for $ t_{249}(0,975) $. I can generate in R in easy way, but in school we will have just tables and in tables are only critical values for degrees of freedom = 100,200,500. How ...
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55 views

The necessity of t-distributions in computing confidence intervals

If the width of a 95% CI can be computed as 1.96 * SEM, where SEM is just SD/sqrt(N), then why is the t-distribution involved in ...
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0answers
125 views

Interpreting the best-fit student t distribution (based on maximum likelihood estimation) for my datasets

I have fitted (through maximum likelihood estimation of parameters) a number of leptokurtic probability density functions (i.e. logistic hyperbolic-secant Laplace log-logistic student's t (with ...
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2answers
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one tail hypothesis testing, Gaussian versus t-distribution

Since the t-distribution represents more uncertainty than the Gaussian distribution I am accustomed to expecting a greater p-...