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Necessary conditions for defining correlation

-1

$begingroup$

Suppose I have a probability space on which I define the space of all random variables. On that given space I then define function F which takes two random variables as arguments and returns a real number between -1 and 1. Let F have such property: [F(X,Y) = 1 iff X = aY + b (almost everywhere) for a>0] and [F(X,Y) = -1 iff X = aY + b (almost everywhere) for a<0]. Have I necessarily defined the normal correlation or there are some other functions with the same properties (if so, what conditions should I add in order to get the normal correlation if it is even possible to get the correlation moving in that direction)?

probability-theory random-variables correlation

share|cite|improve this question

edited Mar 31 at 22:07

graffermed

asked Mar 31 at 17:58

graffermedgraffermed

44

$endgroup$

add a comment | 

-1

$begingroup$

Suppose I have a probability space on which I define the space of all random variables. On that given space I then define function F which takes two random variables as arguments and returns a real number between -1 and 1. Let F have such property: [F(X,Y) = 1 iff X = aY + b (almost everywhere) for a>0] and [F(X,Y) = -1 iff X = aY + b (almost everywhere) for a<0]. Have I necessarily defined the normal correlation or there are some other functions with the same properties (if so, what conditions should I add in order to get the normal correlation if it is even possible to get the correlation moving in that direction)?

probability-theory random-variables correlation

share|cite|improve this question

edited Mar 31 at 22:07

graffermed

asked Mar 31 at 17:58

graffermedgraffermed

44

$endgroup$

add a comment | 

$begingroup$

Suppose I have a probability space on which I define the space of all random variables. On that given space I then define function F which takes two random variables as arguments and returns a real number between -1 and 1. Let F have such property: [F(X,Y) = 1 iff X = aY + b (almost everywhere) for a>0] and [F(X,Y) = -1 iff X = aY + b (almost everywhere) for a<0]. Have I necessarily defined the normal correlation or there are some other functions with the same properties (if so, what conditions should I add in order to get the normal correlation if it is even possible to get the correlation moving in that direction)?

probability-theory random-variables correlation

share|cite|improve this question

edited Mar 31 at 22:07

graffermed

asked Mar 31 at 17:58

graffermedgraffermed

44

$endgroup$

share|cite|improve this question

edited Mar 31 at 22:07

graffermed

asked Mar 31 at 17:58

graffermedgraffermed

44

share|cite|improve this question

edited Mar 31 at 22:07

graffermed

asked Mar 31 at 17:58

graffermedgraffermed

44

asked Mar 31 at 17:58

graffermedgraffermed

44

asked Mar 31 at 17:58

graffermedgraffermed

44