[4 marks] 8. X follows a binomial distribution with pmf: fx (2) - ( x ) D(1 - P)"x x= 0,1, 2,.. otherwise Prove E(X) = np by filling the blanks in the following steps. Proof: E ( Y) = >yfy(x) a Reason: 4=0 In! y = 0 (n - y) !y!P.(1 -p)"-y n n! (n - y)!y - 1)1P"(1 -p)ny b) Reason: n(n - 1)! (n - y)!(y - 1)!P(1-p)ny En ) (n - 1 ) ! (n - y)!(y - 1)!P(1 -p)-y (n - 1)! (n - y)!(y - 1)!P (1 -p)"-y m = n m! (m - x)!(x) P (1-p)m-x m m! = n ro (m - r)!(x)!PP(1 - p)ma m m! = np (m - x)!(x)!P (1 -p)me 2=0 = c) d) Reason