Let E, F, G be events in a sample space S. We say that two events E and F are conditionally independent given G if P (E ∩ F |G) = P (E|G)P (F |G).
(a) If E and F are independent events, are E and F necessarily conditionally independent given G? Provide a proof or counterexample to justify your answer.
(b) If E and F are conditionally independent given G, is it necessarily true that E and F are independent events? Again, provide a proof or counterexample to justify your answer.
Suppose that a die is rolled twice. Let X be the random variable "the minimum value that appears in the two rolls". Let Y be the random variable "the value of the first roll minus the value of the second roll".
(a) What values can X take on? What values can Y take on?
(b) Compute the probability mass function for X. Compute the cumulative distribution
function for X. Graph these functions.
(c) Compute the probability mass function, and the cumulative distribution function for Y . Graph these functions.