Student’s t-Distribution
The student’s^3 t-distribution with ndegrees of freedom is given by the probability
density function
f(x) =
√^1
nπΓ(n+ 1
2)Γ(n
2)(
1 +x^2
n)−(n+1)/ 2
, −∞ < x < ∞ (11 .75)where Γ( ) denotes the gamma function and n = 1, 2 , 3 ,... is a parameter. The
student’s t-distribution has the mean 0 for n > 1 , otherwise the mean is undefined.
Similarly, the variance is given by
nn− 2 for n >^2 , otherwise the variance is undefined.
The cumulative distribution function is given by
F(x) =∫x−∞f(x)dx =∫x−∞√^1
nπΓ(
n+ 1
2)Γ(n
2)(
1 +x^2
n)−(n+1)/ 2
dx (11 .76)The table 11.6 contains values of tα,n which satisfy the equation
∫∞tα,nf(x)dx =α= 1 −F(tα,n ) (11 .77)The normal distribution is related to the student’s t-distribution as follows. If
x and s are the mean and standard deviation associated with a random sample
of size nfrom a normal distribution N(x;μ, σ^2 ), then the quantity (x−μ)
√
ns has a
student-t-distribution with n− 1 degrees of freedom.
The student’s t-distribution is a continuous probability distribution used to esti-
mate the mean of a population where (i) the population has a normal distribution (ii)
the sample size from the population is small and (iii) the standard deviation of the
population is unknown. The table 11.7 gives values for area under this probability
density function.
(^3) Developed by W.S. Gosset who used the name ”Student” as a pseudonym.