Student's t-Distribution

What is it?

The Student’s t-distribution is a Probability distribution widely used in Statistics to estimate population parameters for small sample sizes or unknown Variance.

It’s very similar to the Normal Distribution, with a bell-curve and symmetric shape around 0, but it differs in its heavier tails. This translates to higher kurtosis, meaning that the Variance is higher and extreme values are more probable than the Normal Distribution.

The t-distribution, along with Student’s t-Test, plays a important role for Hypothesis Testing and Statistical Significance.

Student? Weird name.

The Student distribution was developed by [William Gosset], who signed its research papers with the name Student.

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Probability Density Function

The Student t-distribution has the Probability Density Function given by:

$$f(t)=\frac{\Gamma (\frac{v+1}{2})}{\sqrt{\pi v}\ast \Gamma (\frac{v}{2})}{\left(1+\frac{t^2}{v}\right)}^{-(v+1)/2}$$

Where $\Gamma$ is the Gamma function, $v$ is the number of degrees of freedom and $t$ is the Student’s t-Value.

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