1 Suppose that X is generated from a binomial distribution with pmf given by: 3': P(X:m):(n)pm(1p)_ml$:O!1'l"'!nl where n is a positive integer, 0 S p g 1, and for every xed pair n and p the pmf sums to one. Show that the expected value of this binomial random variable is equal 1 to np. (Hint: usethe identity: :3 ( n ) ='n,( n 1 )J. :r :1: 2 Suppose that you have three securities from different industries which are part of the 3&P500 index. Each security has a 7% probability of being removed from the index when this is rebalanced at the end of the year.1 The event of removal for any 1 Rebalancing is the process of realigning the weightings of a portfolio of assets (here the 3&P500 index). Rebalancing involves periodically adding / removing assets from the index on the basis of prespecied criteria. given security is independent of the other securities being removed. What is the probability that zero, one, two, or all three securities are removed from the 3&P500 index at the end of the year? What is the mean number of removals