Events independence

[mchiuminatto]

Hello there.

My question is related to establish random variables independence.

I have variables X and Y and the following joint distribution table:

X\Y	-1	0	1	Total
-1	0.234000	0.003323	0.256152	0.493475
0	0.003756	0.000048	0.003660	0.007464
1	0.255767	0.004093	0.239201	0.499061
Total	0.493523	0.007464	0.499013	1.000000

If X and Y are independent then:

P(X and Y) = P(X)*P(Y)

P(X=1 and PY=1) = P(X=1) * P(Y=1) (1)

In this case:


P(X=1 and PY=1) =0.239201
P(X=1) = 0.499061
P(Y=1) = 0.499013

Replacing values in (1) and rounding to two decimals

0.24 = 0.5*0.5
0.24 = 0.25

Values are very close, so my question is:

Can I consider X and Y independent?
If so, what is the tolerance for the difference to establish that they are independent?


Thank you very much

Regards

Marcello

 

[Steven G]

Of course the answer to your question is no, even if we decided that .24~.25 is good enough. Why? Because you must show that P(X=2 and PY=1) = P(X=2) * P(Y=1) , P(X=0 and PY=1) = P(X=0) * P(Y=1), etc.

Now if any one of the required equation is not valid then you stop and say that X and Y are NOT independent. So your question should be if P(X=1 and Y=1) ~ P(X=1) * P(Y=1) should I say that X and Y are dependent and stop verifying other equations OR should I continue verifying other equations since P(X=1 and Y=1) and P(X=1) * P(Y=1) are close enough.

That really depends. If you know that your given probabilities are exact, then you conclude dependent. If you are told that the given probabilities are exact up to some tolerance then you must decide if P(X=1 and Y=1) and P(X=1) * P(Y=1) are within the allowable tolerance.

 

[Steven G]

I just noticed that you approximated in getting .24 and .25! Maybe if you the used given values you would have equally or within the given tolerance level.

 

[mchiuminatto]

Thank you so much Jomo, it is clear for me now what to do next.

Best regards

Marcello