symmetric random variable
Let be a probability space and a real random variable defined on . is said to be symmetric if has the same distribution function as . A distribution function is said to be symmetric if it is the distribution function of a symmetric random variable.
Remark. By definition, if a random variable is symmetric, then exists (). Furthermore, , so that . Furthermore, let be the distribution function of . If is continuous at , then
so that . This also shows that if has a density function , then .
There are many examples of symmetric random variables, and the most common one being the normal random variables centered at . For any random variable , then the difference of two independent random variables, identically distributed as is symmetric.
|Title||symmetric random variable|
|Date of creation||2013-03-22 16:25:45|
|Last modified on||2013-03-22 16:25:45|
|Last modified by||CWoo (3771)|
|Defines||symmetric distribution function|