The aggregate claim amount is defined as S X₁, where Xį are independent and i=1 identically distributed claim amounts and N is the number of claims. (a) Suppose N follows a negative-binomial distribution with E(N) = k(1 − p)/p and variance Var(N) = k(1 − p)/p². Show that: k(1 − p) (i) E(S) = -m₁ and р k(1 − p) k(1-P)² m², (ii) Var (S) = -M2 + p² - (iii) Skew k(1 − p)m² 3k(1 − p)²m₁m2 p² + + 2k(1 − p)³m³ p³ P
