Which of the following statements are true?多项选择题

(A) The sum rule of probability states that the probability of a variable X, denoted as p(X), can be calculated as the sum of joint probabilities of X and another variable Y across all values of Y.

(B) According to Bayes' theorem, the probability of cancer given a positive test result, p(C = 1|T = 1), can be calculated without considering the overall probability of a positive test result, p(T = 1).

(C) In the context of probabilities, the product rule allows us to express the joint probability of two events X and Y as the product of the conditional probability of Y given X and the probability of X.

(D) If two events are independent, the joint probability of these events equals the product of their individual probabilities, which contradicts the product rule.

(E) Expectations of a function under a probability distribution can be estimated as a finite sum over a set of data points drawn from that distribution, and this approximation becomes exact as the number of data points approaches infinity.