Download 11 Measures with Lebesgue densities

Measures with Lebesguedensities
11
93
Measures with Lebesguedensities
dening measures; the integrands that appear thereby, are called densities. Finally,
2
the normal distribution N (a, σ ) is dened via its density.
The integral can be used for
The following theorem a consequence of the theorem of monotone convergence (which is not reproduced
here) opens the possibility to dene measures by integrals.
11.1 Theorem
Let (Ω, A, µ) be a measure space and f : Ω → R+ a
nonnegative, measurable function,
11.1.1 The function
Z
ν(A) =
f dµ
(A ∈ A)
A
which is dened by ν : A → R is a measure.
11.1.2 If
Z
ν(Ω) =
f dµ = 1 ,
Ω
then ν is a probability measure.
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11.2 Theorem
Let (Ω, A) be a measurable space and µ and ν two
measures on A.
A measurable function f : Ω → R+ such that
Z
(11.2.1)
ν(A) =
f dµ
(A ∈ A)
A
is called a µdensity of ν in symbols ν = f µ.
If (Ω, A) = (Rn , B n ) and µ = λn , then f is called a
Lebesguedensity or λn density.
11.3 Remarks
11.3.1 To have a density means that the values ν(A) for A ∈ A can be represented as
integrals of a real, nonnegative, measurable function f .
To postulate the existence of density is not obvious. Discrete measures, the pointmeasure
µω at ω for example, have no Lebesguedensities.
The assumption of the existence of densities
more specic results in Statistics
11.3.2 entails
Whenmuch
the integral
(for instance) then Z
the general case.
f dµ
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is computed, the integrand f can be altered on
a µnullset, i.e. on a set N ∈ A with µ(N ) =
0 without any inuence on the integral. For
this reason one speaks of a density and not of
the density of a measure.
11.4 Example (standard normal distribution)
The mapping f : R → R+ dened by
(11.4.1)
x2
1
f (x) = √ e− 2
2π
(x ∈ R)
is continuous and therefore due to 9.2.4 measurable.
For f to be a Lebesguedensity of a probability measure P it must hold:
Z +∞
x2
1
√ e− 2 dx = 1
(11.4.2)
P (Ω) =
2π
−∞
due to the required normedness of P ; (11.4.2) holds
according to (10.4).
(11.4.1) denes the standard normal distribution N(0, 1)
with the parameters 0 and 1.
11.5 Denition
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Let a, σ ∈ R, σ > 0.
The probability measure N (a, σ 2 ) which is dened by
the density ga,σ2 : R → R+
(11.5.1)
1
(x − a)2
ga,σ2 (x) = √ exp −
(x ∈ R)
2σ 2
σ 2π
is called the normal distribution with the parameters a and σ 2 . As abbreviation for this normal distribution dened by (11.5.1) we use N(a, σ 2 ).
The graph of the density (11.5.1) is symmetric w.r.t. the
ordinate x = a. At x = a this density attains a maximum and exhibits points of inection at x = a ± σ .
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σ=
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σ=1
σ=2
−4
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Fig. 11.1
The graphs of the density given by (11.5.1) are called
Gaussian bell curves. Obviously, the 'bell curves' are
steep for σ 2 being small and at for σ 2 being large.
Experiment 15.1 shows the generation of N (0, 1) sample realisations that are computed using the Box
Müllermethod. The BoxMüllermethod, that is not
described here, belongs to the topic '15 Random Generators'.
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A detailled presentation can be found in Moeschlin
et al., 'Experimental Stochastics', Berlin, Heidelberg,
New York, 1998.