Understanding Probability Distributions on Subsets: Exploring X and X' Sets

[leonid.ge]

Please help what term to use for the following issue.

Given a set X = { x1, x2, ..., x_n } and a probability distribution on it P (X) = { p (x1), p (x2), ..., p (x_n) }.

Given a division of the set Х on non-overlapping subsets Х1, Х2, ... Х_m, so:
X1 U X2 U ... U X_m = X

Is there a term for the probability distribution on the set of the subsets X' = { X1, X2, ..., X_m }:

P (X') = { p (X1), p (X2), ..., p (Х_m) }, where p (Xi) - the sum of probabilities of all x in Xi?

Thank you in advance.

It seems it is well known issue, for example, say we have a dice with uniform probability 1/6 for each number and we are interested in two events: (A) having 1 or 2 and (B) having 3 or 4 or 5 or 6.
Then p (A) = 2/6 and p (B) = 4/6 and the probability distribution on the set { A, B } is: { 2/6, 4/6 }.
So is there a name for this probability distribution?

 

[tiny-tim]

Welcome to PF!

Given a division of the set Х on non-overlapping subsets Х1, Х2, ... Х_m, so:
X1 U X2 U ... U X_m = X

Is there a term for the probability distribution on the set of the subsets X' = { X1, X2, ..., X_m }:

P (X') = { p (X1), p (X2), ..., p (Х_m) }, where p (Xi) - the sum of probabilities of all x in Xi?

Hi leonid.ge! Welcome to PF!

I think it would probably be called the "induced distribution" … like an induced topology or an induced algebra (but I don't think it comes up often enough for people to want to give it a name ).

 

[leonid.ge]

Hi leonid.ge! Welcome to PF!

I think it would probably be called the "induced distribution" … like an induced topology or an induced algebra (but I don't think it comes up often enough for people to want to give it a name ).

Hi tiny-tim,

Thank you very much!

I write a paper where I always use such an 'induced' distribution, so I need this name.
Actually I even calculate the entropy of this distribution, so perhaps I will use the term 'induced entropy'.

This is what I have found in the Internet concerning the "induced probability", seems this is exactly what I need:

"Simply stated, if a new or random variable is defined in terms of a first random variable, then induced probability is the probability or density of the new random variable that can be found by summation or integration over the appropriate domains of the original random variable" (http://eric.ed.gov/ERICWebPortal/cu...&ERICExtSearch_SearchType_0=no&accno=ED362559).