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
t
f
,
)
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
t
T
P
dt
t
f
dt
t
f
t
UTPT
)
(
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
