Need help with number 3 and 4 please. Thank you!

Exercise 2 Variance of Sum of Random Variables Consider a urn that contains 5 red balls and 6 green balls. Draw 3 balls without replacement from this urn and let X be the number of red balls. 1. What is [(X)? 2. What is Var(X)? 3. Denote Y the number of red balls that you get if you drew 11 balls. What is Var(Y)? How could you have found this result without computing the variance? 4. Assume that the number of balls in the urn grows to innity while the portion of red balls in the urn remains constant and the number of balls you draw remains constant. (In the notations below, N > +oo,K ) +00, such that N / K = p remains constant). What variance do you obtain in this limit? Do you recognize the variance of a classical distribution? Hint: This is an instance of a hypergeometric random variable. Denote N = 11 the total number of balls. K = 5 the number of red balls and n = 3 the number of balls that are drawn from the urn. (Try to express your result rst in terms of N, n, p = K/N and q = 1 p : (N K)/N. Then you can plug the values of N , K , n) The p.m.f. of X is known such that one can use the denition of expectation and variance with respect to the p.m.f. Computing the expectation and the variance can be done more easily by decomposing X in a sum of exchangeable indicator random variables