- #1

WWCY

- 479

- 12

##\textbf{1}##

If I find that some events ##A, B, C## obey the following formula

$$P(A \cap B \cap C ) = P(A)P(B)P(C)$$

it does not necessarily mean that a) they are mutually independent and b) ##A## and ##B## (or any two of the 3 events) are independent unless I explicitly find that all possible subcollections are mutually independent as well. Is this right?

##\textbf{2}##

Similarly, if I find that a set of discrete random variables ##X, Y, Z## obey the following (for specific ##i,j,k##),

$$P(X=x_i \wedge Y = y_j \wedge Z=z_k ) = P(X=x_i)P(Y=y_j)P(Z=z_j)$$

I am not allowed to conclude that a) they are mutually independent and b) any two of the 3 variables are independent unless I explicitly show that ##X,Y,Z## obey the above formula for all ##i,j,k##. Is this right?