Assume we have three independent random variables A, B and C. The distribution for A,B and C are known(In this case just assume they are all beta distribution with different parameters). I want to compare the three random variables.
Is this equation ${\rm Pr}(C > A\ and\ C > B) = {\rm Pr}(C > A){\rm Pr}(C > B)$ true or false?I don't know if $C>A$ and $C>B$ are independent of each other or not. Please let me know how to judge that.
Also, ${\rm Pr}(C > A\ or\ C > B) = {\rm Pr}(C > A) + {\rm Pr}(C > B) - {\rm Pr}(C > A\ and\ C > B)$should always be correct since $C>A$ and $C>B$ are not mutually exclusive. Please let me know if I'm right.
No, $C > A$ and $C > B$ are generally not independent, and so $$Pr(C > A \ \text{and}\ C > B) \ne Pr(C > A) Pr(C > B)$$ Examples are easy (e.g. suppose $A$ and $B$ are almost surely constant).
As for your second question, $Pr(E \ \text{or}\ F) = Pr(E) + Pr(F) - Pr(E \ \text{and}\ F)$ is always true, and does not depend on $E$ and $F$ not being mutually exclusive. Of course if they do happen to be mutually exclusive, $Pr(E \ \text{and}\ F) = 0$.