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# What is T-Distribution?

## What is T-Distribution?

The (student’s) t-distribution is closely related to the normal, but it has heavier tails. The student’s t distribution was originally developed for testing hypotheses using small samples.
A Student’s t is a one-parameter distribution. Denoted by v, this parameter is also called the degrees of freedom parameter. While it affects many aspects of the distribution, the most significant effect is on the shape of the tails.

## Example of T-Distribution:

A Student’s t is the distribution of:

$Y\, =\, \frac{Z}{\sqrt{\frac{W}{r}}}$

where, Z is a standard normal, W is a chi-squared random variable, and Z and W are independent. Dividing a standard normal by another random variable produces heavier tails than the standard normal. This is true for all values of ν, although a student’s t converges to a standard normal as ν→∞.

If Y~ tv, then the mean is:

E[Y]=0

and the variance is:

V[Y] = $\frac{\nu }{\nu -2}$

## Why is t-distribution important?

The t-distribution is most useful for small sample sizes, when the population standard deviation is not known, or both. The t-distribution becomes more similar to a normal distribution as the sample size increases.

Owais Siddiqui
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