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How to find the expected value of the ratio?
Had be given the ratio constraint, we could employ a standard rejection sampling scheme, independently simulating and then rejecting all pairs violating the condition: So the problem is to come up with an estimator of the ratio ; one such estimator could be the expected value of the ratio of .
Is the expectation of a random variable a linear operator?
In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations. 1. If g(x) ≥ h(x) for all x ∈ R, then E[g(X)] ≥ E[h(X)].
Is the mean and variance of a random pair the unknown ratio?
We are only given the marginal distributions (or rather the mean and variance) of , not the ratio constraint that the data obeyed Simulating from the marginals will result in pairs of that marginally have the “correct” mean and variance, yet many of these pairs may violate the unknown ratio constraint.
How to find the estimator of the ratio?
So the problem is to come up with an estimator of the ratio ; one such estimator could be the expected value of the ratio of . Therefore, we need to find a way to compute the estimator using only the marginal statistics provided. Firing off the calculus cannon the initial order approximations are:
Which is the best rule for correlation in random variables?
If the random variables are correlated then this should yield a better result, on the average, than just guessing. We are encouraged to select a linear rule when we note that the sample points tend to fall about a sloping line. Yˆ =aX +b. where a and b are parameters to be chosen to provide the best results.
Can you write e ( xy ) as an expectation?
If X and Y are independent, then E(XY) = E(X)E(Y). However, the converse is not generally true: it is possible for E(XY) = E(X)E(Y) even though X and Y are dependent. Probability as an Expectation Let A be any event. We can write P(A) as an expectation, as follows.