independent identically distributed
Two random variables X and Y are said to be identically distributed if they are defined on the same probability space
(Ω,ℱ,P), and the distribution function
FX of X and the distribution function FY of Y are the same: FX=FY. When X and Y are identically distributed, we write Xd=Y.
A set of random variables Xi, i in some index set I, is identically distributed if Xid=Xj for every pair i,j∈I.
A collection of random variables Xi (i∈I) is said to be independent identically distributed, if the Xi’s are identically distributed, and mutually independent
(http://planetmath.org/Independent) (every finite subfamily of Xi is independent). This is often abbreviated as iid.
For example, the interarrival times Ti of a Poisson process of rate λ are independent and each have an exponential distribution with mean 1/λ, so the Ti are independent identically distributed random variables.
Many other examples are found in statistics, where individual data points are often assumed to realizations of iid random variables.
Title | independent identically distributed |
---|---|
Canonical name | IndependentIdenticallyDistributed |
Date of creation | 2013-03-22 14:27:29 |
Last modified on | 2013-03-22 14:27:29 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 8 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 60-00 |
Synonym | iid |
Synonym | independent and identically distributed |
Defines | identically distributed |