(Lamperti) An urn contains exactly 5000 balls, of which an unknown number [math]X[/math] are white and the rest red, where [math]X[/math] is a random variable with a probability distribution on the integers 0, 1, 2, ..., 5000.

• Suppose we know that [math]E(X) = \mu[/math]. 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?
• 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
  [[math]] P({{\rm white},{\rm white}}) = P({{\rm white}})^2\ ? [[/math]]
• Suppose the variance of [math]X[/math] is [math]\sigma^2[/math]. What is the probability of drawing two white balls in part (b)?

Notes