t-distribution

Saturday, 17 August 2019

5:16 PM

He showed that, in a normal population, the exact distribution of T is the so-called t-distribution with n — 1 degrees of freedom: T tn-l This distribution is now referred to as Student's t-distribution (which might otherwise have been Gosset's t-distribution).

 

A random variable, say T, is said to follow the Student's t-distribution with v degrees of freedom, i.e. Its probability density function is given by 2) t2 2 f(t) — v/DRr (;) for some integer v. Note: the Gamma function is given by r(y) - for y > 0 It can be shown that r (y) — (y — 1) x r (y — 1), so that, if y is a positive integer n,

 

and var(T) - (for v > 2)

 

The Student's t distribution is similar in shape to the standard normal distribution in that both densities are symmetric, unimodal and bell-shaped, and the maximum value is reached at 0. However, the Student's t distribution has heavier tails than the normal there is more probability to find the random variable T 'far away' from 0 than there is for Z

 

The Student's t-distribution: quantiles Similarly to what we did for the Normal distribution, we can define the quantiles of any Student's t-distribution: tv—distributm Let tv-a be the value such that > tv;o) - for T tv Like the standard normal distribution, the symmetry of any tv-distribution implies that tv;l-a ¯ 6 6 -2 2 is also referred to as t critical value.

 

t-confidence interval on the mean of a normal distribution -+ if R and s are the sample mean and sample standard deviation of an observed random sample of size n from a normal distribution, a confidence interval of level 100 x (1 — n)0/o for is given by s s This confidence interval is sometimes called t-confidence interval, as ] (z-confidence interval) opposed to i — + Zl—a/2Vh Because tn_l has heavier tails than M(O, 1), > 4 _ 0/2, Vn —+ this reflects the extra variability introduced by the estimation of a (less accuracy)

 

 

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