(adsbygoogle = window.adsbygoogle || []).push({}); "Factorizing" random variables

Suppose we have a (discrete) random variable X. Consider a random variable Y "equivalent" to X if there are functions f, g such that X = f(Y) and Y = g(X). Among other things, this implies H(Y) = H(X).

Y = Y_{1}x Y_{2}x ... x Y_{n}, where x is the cartesian product, is an "independent factorization" of X if Y is equivalent to X and the Y_{i}are mutually independent random variables (the factors).

So for example, if X is uniformly distributed over {0,1,2,3}, then a factorization is Y = B_{1}x B_{2}where the B_{i}are uniformly distributed over {0,1} and mutually independent. f(x) = (x mod 2, floor(x / 2)), and g((b1,b2)) = b1 + 2*b2.

In this context, a random variable Y is "one" if H(Y) = 0, that is if Y takes one value with probability 1. A random variable Y is irreducible if it has no factors other than one and itself.

A factorization of a random variable is irreducible if all factors in the factorization are irreducible, and are not one.

The problem is then, given a discrete random variable, what are its factorizations? Are irreducible factorizations of discrete random variables unique up to order of the factors and equivalence (as defined above) of the factors? Are irreducible elements also prime?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Factorizing random variables

Loading...

Similar Threads for Factorizing random variables |
---|

A Why does Polychoric Reduce to two Factors? |

I Regression on extracted factors |

I Order of "Extracted Factors" in SPSS Factor Analysis |

A References Request: Factor Analysis on Non-Continuous Variables |

A Advanced Data Fitting - More than Simple Regressions |

**Physics Forums | Science Articles, Homework Help, Discussion**