An urn contains exactly 5000 balls, of which an unknown number X are white and the rest red, where X is a random variable with a probability distribution on the integers 0, 1, 2, . . . , 5000.
(a) Suppose we know that E(X) = µ. Show that this is enough to allow us to calculate the probability that a ball drawn at random from the urn will be white. What is this probability?
(b) We draw a ball from the urn, examine its color, replace it, and then draw another. Under what conditions, if any, are the results of the two drawings independent; that is, does P (white, white) = P (white)2 ?
(c) Suppose the variance of X is σ2. What is the probability of drawing two white balls in part (b)?