The Distribution Function
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The Distribution Function
There exist dierent possibilities to dene a probability measure.
One such possibility is the representation by the distribution
function for probability measures on B over R which was
introduced by A. von Mises.
The extension of the representation concept to the Rn is rather
not handy.
A possible way to dene a probability measure on
(R, B) uses the distribution function. Such a distribution function may also be introduced over the Rn ,
but in this case the concept of the distribution function
is not very handy.
8.1 Distribution function
8.1.1 Let P be a probability measure on B (over R),
then the function
(8.1.1.1) FP (x) := P ((−∞, x])
8.1.2
is called distribution function
FP has the following properties
(8.1.2.1)
FP is
(x ∈ R)
of P .
isotone
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(8.1.2.2)
FP is (as a consequence of the right closed intervals
(−∞, x] in (8.1.1.1))
(8.1.2.3)
(8.1.2.4)
8.1.3
rightside continuous.
lim FP (x) = 0
x→−∞
lim FP (x) = 1
x→+∞
Any function F , satisfying (8.1.2.1) (8.1.2.4)
denes a probability measure PF on B with
(8.1.3.1)
PF ((a, b]) = F (b) − F (a) ,
F being its distribution function. Note, the
distribution function is a representation concept for probability measures P on (R, B).
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8.2 Examples of distribution functions
8.2.1 Distribution function of a discrete probability
measure on (R, B)
1
....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ..............
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b
8.2.2
b
b
b
b
−1 0 1 2 3 4 5 6
Denition (8.1.1.1) is based on the interval
(−∞, x]; therefore the intervals of the graph
given above are left closed and right open !
Distribution function of the exponential distribution
The function F : R → R
(
1 − e−αx (x ≥ 0)
F (x) :=
0
(x < 0)
satises the conditions (8.1.2.1) (8.1.2.4) of a
distribution function; the assigned probability
measure is called the exponential distribution .
The Distribution Function
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....... ....... ....... ....... ....... ....... .......................................................................... ....... ....... .......
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−1 0 1 2 3 4 5 6
The (generalized) inverse distribution function proves
useful for the generation of realizations according
to a probability measure P .
8.3 Denition (generalized inverse distribution
function) (facultative)
Let P be a probability measure on (R, B) with distribution function F .
The generalized inverse distribution function F inv :=
FPinv of P
F inv : (0; 1) → R
is dened by
F inv (u) := inf{x ∈ R| F (x) ≥ u},
u ∈ (0; 1) .
We renounce on a discussion of F inv and mention, that
its graph may be generated by reexion of F of P at
the 45◦ line of the rst quadrant.
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We explain the construction principle of F inv with the
help of a gure.
8.4 Construction principle of the generalized inverse distribution function (facultative)
Let be given the graph of the distribution function F
of P :
R
1 6
u3
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F
- R
0 F
inv
(u1 )
F
inv
(u2 ) F
inv
(u3 )
Fig. 8.1
To the elements u1 , u2 , u3 ∈ (0; 1) (vertical axis), the
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images F inv (u1 ), F inv (u2 ), F inv (u3 ) (horizontal axis)
are assigned according to g. 8.1.
The following fact is meaningful.
8.5 Theorem (facultative)
Let P be a probability measure on (R, B) with F inv
as (generalized) inverse distribution function and let
λ(0,1) be the BL measure restricted to (0; 1) (uniform
distribution on) (0; 1).
Then we have
(λ(0,1) )F inv = P ,
i.e. the
image measure of λ(0,1) under F inv equals
the probability measure P .