Four students enter the lift of the Five-storey building. Assume…
Question Answered step-by-step Four students enter the lift of the Five-storey building. Assume… Four students enter the lift of the Five-storey building. Assume that each of them exits uniformlyat random at any of five levels and independently of each other. In this question we study therandom variable Z, which is the total number of lift stops (you may want to re-use somecalculations 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 therandom 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 definition of expectation. Your answershould be in agreement with (b) and (d). Please solve (f) and (g) Math Statistics and Probability MAT 9004 Share QuestionEmailCopy link Comments (0)
