HP 50G, User Manual

Page: 561 / 887

Page 17-12
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,
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:
Θ
P(X<a) = 1 - UTPC(
ν
,a)
Θ
P(a<X<b) = P(X<b) - P(X<a) = 1 - UTPC(
ν
,b) - (1 - UTPC(
ν
,a)) =
UTPC(
ν
,a) - UTPC(
ν
,b)
Θ
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
∫ ∫
∞ −
∞
≤ − = − = =
t
t
x X P dx x f dx x f x UTPC
) ( 1 ) ( 1 ) ( ) , (
ν
)
2
(
1
2 2
) 1 ( )
2
( )
2
(
) ( )
2
(
) (
D N
N N
D
F N D N
F
D
N D N
x f
ν ν
ν ν
ν
ν ν ν
ν
ν ν ν
+
−
⋅
− ⋅ Γ ⋅ Γ
⋅ ⋅
+
Γ
=