distribution function (original) (raw)
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Let F:ℝ→ℝ. Then F is a distribution function
if
- F is nondecreasing,
- limx→-∞F(x)=0, and limx→∞F(x)=1.
As an example, suppose that Ω=ℝ and that ℬis the σ-algebra of Borel subsets of ℝ. Let P be a probability measure
on (Ω,ℬ). Define F by
This particular F is called the distribution function of P. It is easy to verify that 1,2, and 3 hold for this F.
In fact, every distribution function is the distribution function of some probability measure on the Borel subsets of ℝ. To see this, suppose that F is a distribution function. We can define P on a single half-open interval by
and extend P to unions of disjoint intervals by
| P(∪i=1∞(ai,bi])=∑i=1∞P((ai,bi]). |
|---|
and then further extend P to all the Borel subsets of ℝ. It is clear that the distribution function of P is F.
0.1 Random Variables
Suppose that (Ω,ℬ,P) is a probability space andX:Ω→ℝ is a random variable. Then there is an_induced_ probability measure PX on ℝ defined as follows:
for every Borel subset E of ℝ. PX is called the_distribution_ of X. The _distribution function_of X is
The distribution function of X is also known as the law of X. Claim: FX = the distribution function of PX.
| FX(x) | = | P(ω|X(ω)≤x) |
|---|---|---|
| = | P(X-1((-∞,x]) | |
| = | PX((-∞,x]) | |
| = | F(x). |
0.2 Density Functions
Suppose that f:ℝ→ℝ is a nonnegative function such that
Then one can define F:ℝ→ℝ by
Then it is clear that F satisfies the conditions 1,2,and 3 so Fis a distribution function. The function f is called a density functionfor the distribution F.
If X is a discrete random variable with density function f and distribution function F then