Four students enter the lift of the Five-storey building. Assume that each of them exits uniformly at random at any of five levels and independently of each other. In this question we study the random variable Z, which is the total number of lift stops (you may want to re-use some calculations from Question 3 but then you need to explain the connection). (a) Describe the sample space for this random process. (b) Find the probability that the lift stops at a fixed level i 2 f1; 2; 3; 4; 5g. Let Xi be the random variable that equals 1 if the lift stops at level i and 0, otherwise. Compute EXi. (c) Express Z in terms of X1; : : : ;X5. Find EZ using the linearity of the expectation. [2] (d) Find the probability that the lift stops at both levels i and j for i; j 2 f1; 2; 3; 4; 5g. [2] Compute EXiXj . (e) Are the variables X1 and X2 independent? Justify your answer. [1] (f) Compute EZ2 using the formula (X1 +.....+ X5)^2 =EXiXj (where the sum is over all ordered pairs (i; j) of numbers from f1; 2; 3; 4; 5g and the linearity of the expectation. Find the variance VarZ. (g) Find the distribution of Z. That is, determine the probabilities of events Z = i for each [8] i = 1; : : : ; 4. Compute EZ and EZ2 directly by the denition of expectation. Your answer should be in agreement with (b) and (d).