only question 6 is needed to be done.URGENT! MATH 464 H

only question 6 is needed to be done.URGENT!
MATH 464 HOMEWORK 5 SPRING 2013 The following assignment is to be turned in on Thursday February 21 2013. 1. Let x; y 2 R and take n 2 an integer. Prove that (x + y)n =Xn k=0nkxkyn??k using an induction argument. 2. Let X be a binomial random variable with parameters n and p. Find the mean and variance of X. Hint: Sometimes it is easier to calculate E(X(X ?? 1)) = E(X2) ?? E(X) rather than E(X2) directly. 3. Let X be a geometric random variable with parameter p > 0. Find the mean and variance of X. Hint: By geometric series we know that 1Xn=0 rn = 1 1 ?? r for any real number r with jrj 0. Let m and n be non-negative integers. Show that P(X > n + mjX > m) = P(X > n) : This shows that X has the lack of memory property. 6. Let X be the number of eggs laid by an insect. Suppose that X is a Poisson random variable with parameter > 0. Suppose that each egg produces an insect with probability 0