Page 17-12
0
,
0
,
)
2
(
2
1
)
(
2
1
2
2
>
>
⋅
⋅
Γ
⋅
=
−
−
x
e
x
x
f
x
ν
ν
ν
ν
The calculator provides for values of the upper-tail (cumulative) distribution
function for the
χ
2
-distribution using [UTPC] given the value of x and the
parameter
ν
. The definition of this function is, therefore,
∫
∫
∞
−
∞
≤
−
=
−
=
=
t
t
x
X
P
dx
x
f
dx
x
f
x
UTPC
)
(
1
)
(
1
)
(
)
,
(
ν
To use this function, we need the degrees of freedom,
ν
, and the value of the
chi-square variable, x, i.e., UTPC(
ν
,x). For example, UTPC(5, 2.5) =
0.776495…
Different probability calculations for the Chi-squared distribution can be
defined using the function UTPC, as follows:
•
P(X<a) = 1 - UTPC(
ν
,a)
•
P(a<X<b) = P(X<b) - P(X<a) = 1 - UTPC(
ν
,b) - (1 - UTPC(
ν
,a)) =
UTPC(
ν
,a) - UTPC(
ν
,b)
•
P(X>c) = UTPC(
ν
,c)
Examples: Given
ν
= 6, determine:
P(X<5.32) = 1-UTPC(6,5.32) = 0.4965..
P(1.2<X<10.5) = UTPC(6,1.2)-UTPC(6,10.5) = 0.8717…
P(X> 20) = UTPC(6,20) = 2.769..E-3
The F distribution
The F distribution has two parameters
ν
N = numerator degrees of freedom,
and
ν
D = denominator degrees of freedom. The probability distribution
function (pdf) is given by
