(Lamperti) 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)?