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Let . Then is a distribution function if
is continuous from the right,
, and .
This particular is called the distribution function of . It is easy to verify that 1,2, and 3 hold for this .
In fact, every distribution function is the distribution function of some probability measure on the Borel subsets of . To see this, suppose that is a distribution function. We can define on a single half-open interval by
and extend to unions of disjoint intervals by
and then further extend to all the Borel subsets of . It is clear that the distribution function of is .
0.1 Random Variables
for every Borel subset of . is called the distribution of . The distribution function of is
The distribution function of is also known as the law of . Claim: = the distribution function of .
0.2 Density Functions
Suppose that is a nonnegative function such that
Then one can define by
Then it is clear that satisfies the conditions 1,2,and 3 so is a distribution function. The function is called a density function for the distribution .
If is a discrete random variable with density function and distribution function then
|Date of creation||2013-03-22 13:02:51|
|Last modified on||2013-03-22 13:02:51|
|Last modified by||Mathprof (13753)|
|Synonym||cumulative distribution function|
|Defines||law of a random variable|