A tail probability is the probability that a random variable lies beyond a specified threshold. For a real random
variable 
with distribution function 
, the strict upper-tail probability is
 |
(1)
|
As a function of 
, this is the survival
function. The lower-tail probability is 
(NIST).
Including the threshold in the upper tail changes the result by its point probability:
 |
(2)
|
where 
is the left-hand limit of 
at 
. If 
has a probability
density function 
, both upper-tail conventions
agree and
 |
(3)
|
For a random variable with a discrete distribution, a sum of point probabilities replaces the integral, with the threshold included or excluded as specified.
In hypothesis testing, a P-value uses the tail region determined by the test statistic and the alternative hypothesis. It can involve one or both tails rather than always being the upper-tail probability (Orloff and Bloom 2022, pp. 3-4).
