Page 17-11
The Student-t distribution
The Student-t, or simply, the t-, distribution has one parameter ν, known as the
degrees of freedom of the distribution. The probability distribution function
(pdf) is given by
∞<<−∞+⋅
⋅Γ
+
Γ
=
+
−
t
t
tf ,)1(
)
2
(
)
2
1
(
)(
2
1
2
ν
ν
πν
ν
ν
where Γ(α) = (α-1)! is the GAMMA function defined in Chapter 3.
The calculator provides for values of the upper-tail (cumulative) distribution
function for the t-distribution, function UTPT, given the parameter ν and the
value of t, i.e., UTPT(ν,t). The definition of this function is, therefore,
∫∫
∞−
∞
≤−=−==
t
t
tTPdttfdttftUTPT )(1)(1)(),(ν
For example, UTPT(5,2.5) = 2.7245…E-2. Other probability calculations for
the t-distribution can be defined using the function UTPT, as follows:
• P(T<a) = 1 - UTPT(ν,a)
• P(a<T<b) = P(T<b) - P(T<a) = 1 - UTPT(ν,b) - (1 - UTPT(ν,a)) =
UTPT(ν,a) - UTPT(ν,b)
• P(T>c) = UTPT(ν,c)
Examples: Given ν = 12, determine:
P(T<0.5) = 1-UTPT(12,0.5) = 0.68694..
P(-0.5<T<0.5) = UTPT(12,-0.5)-UTPT(12,0.5) = 0.3738…
P(T> -1.2) = UTPT(12,-1.2) = 0.8733…
The Chi-square distribution
The Chi-square (χ
2
) distribution has one parameter ν, known as the degrees of
freedom. The probability distribution function (pdf) is given by
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