HP 49g+ - Manual (Page 562)

Page 17-13
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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
=
The calculator provides for values of the upper-tail (cumulative) distribution
function for the F distribution, function UTPF, given the parameters
νN and νD,
and the value of F. The definition of this function is, therefore,
∫ ∫
∞ −
∞
≤ ℑ − = − = =
t
t
F P dF F f dF F f F D N UTPF
) ( 1 ) ( 1 ) ( ) , , ( ν ν
For example, to calculate UTPF(10,5, 2.5) = 0.161834…
Different probability calculations for the F distribution can be defined using the
function UTPF, as follows:
• P(F<a) = 1 - UTPF(νN, νD,a)
• P(a<F<b) = P(F<b) - P(F<a) = 1 -UTPF(νN, νD,b)- (1 - UTPF(νN, νD,a))
= UTPF( νN, νD,a) - UTPF(νN, νD,b)
• P(F>c) = UTPF(νN, νD,a)
Example: Given νN = 10, νD = 5, find:
P(F<2) = 1-UTPF(10,5,2) = 0.7700…
P(5<F<10) = UTPF(10,5,5) – UTPF(10,5,10) = 3.4693..E-2
P(F>5) = UTPF(10,5,5) = 4.4808..E-2
Inverse cumulative distribution functions
For a continuous random variable X with cumulative density function (cdf) F(x)
= P(X<x) = p, to calculate the inverse cumulative distribution function we need
to find the value of x, such that x = F
-1 (p). This value is relatively simple to
find for the cases of the exponential and Weibull distributions since their cdf’s
have a closed form expression: