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probability distribution function∗
Mathprof†
2013-03-21 14:03:39
1
Definition
Let (Ω, B, µ) be a measure space. A probability distribution function on Ω is a
function f : Ω −→ R such that:
1. f is µ-measurable
2. f is nonnegative µ-almost everywhere.
3. f satisfies the equation
Z
f (x) dµ = 1
Ω
The main feature of a probability distribution function is that it induces a
probability measure P on the measure space (Ω, B), given by
Z
Z
P (A) :=
f (x) dµ =
1A f (x) dµ,
A
Ω
for all A ∈ B. The measure P is called the associated probability measure of
f . Note that P and µ are different measures, even though they both share the
same underlying measurable space (Ω, B).
2
Examples
2.1
Discrete case
Let I be a countable set, and impose the counting measure on I (µ(A) := |A|,
the cardinality of A, for any subset A ⊂ I). A probability distribution function
on I is then a nonnegative function f : I −→ R satisfying the equation
X
f (i) = 1.
i∈I
∗ hProbabilityDistributionFunctioni
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1
One example is the Poisson distribution Pr on N (for any real number r),
which is given by
ri
Pr (i) := e−r
i!
for any i ∈ N.
Given any probability space (Ω, B, µ) and any random variable X : Ω −→ I,
we can form a distribution function on I by taking f (i) := µ({X = i}). The
resulting function is called the distribution of X on I.
2.2
Continuous case
Suppose (Ω, B, µ) equals (R, Bλ , λ), the real numbers equipped with Lebesgue
measure. Then a probability distribution function f : R −→ R is simply a
measurable, nonnegative almost everywhere function such that
Z ∞
f (x) dx = 1.
−∞
The associated measure has Radon–Nikodym derivative with respect to λ equal
to f :
dP
= f.
dλ
One defines the cumulative distribution function F of f by the formula
Z x
F (x) := P ({X ≤ x}) =
f (t) dt,
−∞
for all x ∈ R. A well known example of a probability distribution function on
R is the Gaussian distribution, or normal distribution
f (x) :=
2
2
1
√ e−(x−m) /2σ .
σ 2π
2