From the exercise we can conclude that c n 1 therefore we

Question

Question: 1. Use partial derivatives of F(α, β) to show that E(X²Y²) = 2σ₁²σ₂²ρ².

2. Show that E(X²Y²) = E(X²)E(Y²)+2E(XY)².

3. Use integration by parts or the Beta function to show that

From the exercise we can conclude that c = N + 1; therefore we have the pdf P(x|D). Now we want to estimate x itself. One way to do this is to calculate the expected value of this pdf, which, according to the exercise, is (n + 1)/(N + 2). So even though we do not know x, we can reasonably say (n + 1)/(N + 2) is the probability that the next ball will end up in region A, given the behavior of the previous N balls. There is a second way to estimate x; we can find the value of x for which the pdf reaches its maximum. A quick calculation shows this value to be n/N. This estimate of x is not the same as the one we calculated using the expected value but they are close for large N. What is controversial here is the decision to treat the positioning of the line as a random act, whereby x becomes a random variable, as well as the selection of a specific pdf to govern the random variable x. Even if x were a random variable, we do not necessarily know its pdf. Bayes takes the pdf to be uniform over [0, 1], more as an expression of ignorance than of knowledge. It is this broader use of prior probabilities that is generally known as Bayesian methods and not the use of Bayes' Rule itself.