Please give workings

Question 7: We have a fair, 6-sided die that has 1 on two sides, 2 on two sides and 3 on two sides. Thus each time we roll the die we get the values 1, 2, or 3 with equal probability. We will keep rolling this die until the sum of all the values seen so far is > n. Let X, be the value of the i-th roll, let Sn be the final sum, and let Yn be the number of dice rolls. Thus Sn = Xi+ X2+ ... + Xy, and Xy, is the first die roll such that X1 + X2+ ... + Xy, >n. We wish to determine E(Sn). Wald's Identity tells us that, since Yn is a stopping time, that E(Sn) = E(X1) . E(Yn), so we are left with determining E(X1) and E( Yr). a) Determine E(X1 ). b) Determine E(Yn) for n = 1, n = 2, and n = 3. c) Prove that Xi and Yn are NOT independent random variables. d) We are going to find a recursive expression for E( Yn), n 2 3. That is, we will define E(Yn) in terms of E( Yn-1), E(Yn-2), etc. We will use the law of total probability, which states that if the events X1 = 1, X1 = 2, X1 = 3 partition the sample space, then we can express E( Yn) as E ( Y n ) = > E ( Ynl X1 = 1) . Pr (X1 = i). i) Explain in one or two sentences why events X1 = 1, X1 = 2, X1 = 3 partition the sample space. ii) Assume that X1 = 1, that is, the first roll was equal to 1. Express Yn in terms of Yn-1, where Yn-1 is the number of rolls until X2 + .. . + Xy,, sums to n - 1 or more. Find similar expressions for Yn when X1 = 2 and for when X1 = 3. iii) Substitute the above expressions in ii) for E(Y, X, = 1), E(Y, X, = 2) and E(Yn X1 = 3) in equation 1 to find a recursive expression for E(Yn). results and Wald's identity to determine E(S4)