Suppose X and Y are two real random variables. You observe that X = x but don't know the value that Y has taken, and would like to decide with minimum probability of error whether Y is greater than x or less than x. (You can assume that the joint, conditional, and marginal probability density functions of X and Y are continuous, i.e., have no jumps or delta functions.)

(a) Specify the appropriate decision rule for the case where X and Y are independent. (You should find that your answer involves one or more of the following numbers associated with a PDF:

(i) the mean or expected value;

(ii) the median (which is the point where the cumulative distribution function takes the value 0.5, i.e., the probability mass above the median equals the probability mass below it); and

(iii) the mode or modes (which are the points at which the PDF takes its maximum value).)

(b) Specify the appropriate decision rule for the case where X and Y are not independent.