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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28 views

Confidence interval for the mean

If we want to create a confidence interval to estimate the mean of a specific population and we want the margin of error not to exceed $10$ at confidence level of $90\%$ given that some study suggests ...
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47 views

Noncentral t-distribution — relationship to shifted/scaled normal distribution

Let $x$ be 100 random samples from a $N(10,4)$ distribution. Suppose that I want to calculate the likelihood of these data, given my knowledge that $\mu=10,\sigma=4$. For the normal distribution, this ...
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Is there an algebra error in this derivation in a Coursera class of the posterior of a Normal with a Student-distributed mean?

This is a screenshot from Coursera's class "Bayesian Statistics: Techniques and Models", Week 1, "Non-conjugate models" lecture (any one can audit the class and access the materials for free): This ...
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1answer
40 views

estimate the parameters of t-distribution and fitting with MLE

Here is Fitting the t-Distribution by Maximum Likelihood t-method in book Statistics and Data Analysis for Financial Engineering with R examples page 113 and 168. ...
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45 views

how to understand Bootstrap t - method

Here is the definition of bootstrap t-method in book Statistics and Data Analysis for Financial Engineering with R examples page ...
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1answer
43 views

Fitting a multivariate T Copula to three variables in R

I have three variables with correlation coefficient of respectively 0.3; -0.2 and 0.1. I want to fit a T copula but when I use the fitCopula function it gives me a single value for rho, which leads to ...
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1answer
16 views

GARCH (1,1) - stationarity in case of insignificant alpha?

the questions are about GARCH-t (1,1) [t-distribution]. The first question in GARCH-t (1,1) model, the alpha (ARCH) is insignificant. How to rewrite the model? The second one, in case of ...
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37 views

Normal versus student-t distribution

I estimated two GARCH models, one with the Normal distribution and one with the Student-t distribution. The conditional volatility shows less noise for the model based on the Student-t distribution. ...
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Multivariate T distribution?

Suppose $(Y_1,Y_2)^T \sim N((0,0)^T,\Sigma)$, where $$ \Sigma = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}. $$ $T_1= \frac{\bar{Y_1}}...
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Why we don’t make use of the t-distribution for constructing a confidence interval for a proportion?

To calculate the confidence-interval (CI) for mean with unknown population standard deviation (sd) we estimate the population standard deviation by employing the t-distribution. Notably, $CI=\bar{X} \...
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13 views

Tcdf and invT with incomplete beta [duplicate]

I am working in python. I have a function for incomplete beta. Iow would I calculate tCDF and invT using this? the incomplete beta function that I have is this: ...
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8 views

Comparison between Bivariate Student t with dependence matrix and without dependence matrix

Let $\boldsymbol{Y} = (Y_1, Y_2)^T $ be a $n \times 2$ random vector and $\boldsymbol{X}_1, \boldsymbol{X_2}$ are $n \times p$ regressors matrices. Finaly, $\Sigma^{1/2} = \Sigma \times \Sigma$, where ...
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57 views

Calculating confidence interval for RMSE

I'm reading a book on machine learning where the author uses the Random Forest Regression model to fit a dataset. The confidence interval for the root mean squared error is then computed using the ...
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1answer
84 views

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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31 views

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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9 views

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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25 views

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
127 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
30 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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83 views

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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1answer
25 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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1answer
192 views

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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132 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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1answer
131 views

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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1answer
60 views

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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223 views

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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1answer
108 views

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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0answers
31 views

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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23 views

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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31 views

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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30 views

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
98 views

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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14 views

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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1answer
111 views

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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1answer
285 views

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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38 views

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
79 views

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
97 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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1answer
53 views

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
39 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
139 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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1answer
101 views

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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1answer
71 views

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
40 views

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
268 views

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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25 views

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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86 views

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
1k views

Simulation of t copula in Python [closed]

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 ...