Tutorial 1: try, scan and upload these exercises to Blackboard before attending your tutorial in Week 3. Those submitted by the notified deadline will be scored and contribute towards the Tutorial Assignment component of your final mark. 1. Prove, using induction, that 1 x 2 + 2 x 3 + . . . + n(n + 1) _ "(n+1)(n+2) 3 2. Differentiate the following functions with respect to r: (a) S(x) = (ar +6)2 [try this with and without expanding...] (b) /(x) = (ar + b)" (c) /(x) = earth (d) /(x) = e(artb)" (e) /(x) = 1 -e-(athx) ([) S(x) = In(a + br) (g) /(x) = In[(a + br)"] (h) /(x) = ptem +gems (i) S(x) = Eng.(:) 3. Use the product rule (fg)' = 'g + /g', and the chain rule (S(9))' = g'f'(g) to prove the quotient rule (f/g)' = (S'g - fg')/g2. [Hint: f/g =/(g-1)] 4. A random variable X has probability function /(x) = cr, = = 1, 2, ..., n. (a) Determine the constant c. (b) Determine E(X). 5. The kth central moment of the random variable X is defined to be /x = E{(X-/)*} where u = E(X). Derive the form of the second, third and fourth central moments (i.e. k = 2,3, 4 respectively) in terms of the non-central moments, * = E(X*)