# 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 ...
118 views

### 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 ...
1 vote
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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 ...
26 views

### 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]...
788 views

### 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 ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 $n$ ...
35 views

### 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 ...
461 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 ...
1 vote
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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,...
63 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 ...
521 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 ...