IE2100: Question 1 Let X, Y be Continuous RVs with the Following Joint PDF: Probability Models and Applications Assignment, NUS, Singapore

Question 1. Let X, Y be continuous RVs with the following joint PDF:

(a) (6 pts) Compute the marginal PDFs fX(x), fY (y).
(b) (3 pts) Are X, Y independent? Explain your answer.
(c) (3 pts) Compute the conditional PDF fY |X(y|x).
(d) (6 pts) Compute E[Y |X = x].
(e) (2 pt) Express E[Y |X].

Question 2. There are two urns containing dies. A die can be either

• a fair dice, for which the shown value X when rolled follows Unif({1, 2, . . . , 6}), i.e. Pr(X = i) = 1/6 for all i ∈ {1, 2, . . . , 6}, or
• a loaded dice, for which the shown value X when rolled follows the distribution:
  Pr(X = 1) = Pr(X = 2) = Pr(X = 3) = 1/9,
  Pr(X = 4) = Pr(X = 5) = Pr(X = 6) = 2/9.
  Urn I contains 3 fair dies and 5 loaded dies, while Urn II contains 4 fair dies and 3 loaded dies. Consider the following process:

1. You draw a die from Urn I uniformly at random, and put it in Urn II.
2. Then, you draw a die from Urn II uniformly at random and roll it. Denote the face value it shows as Y, which is an RV.
(a) (10 pts) Compute Pr(Y = 2).
(b) (10 pts) Compute E[Y ].

Question 3. (a) (7 pts) Consider N iid RVs X1, . . . , XN , where X1 ∼ Poi(4). Let m ∈ [0, 4N]. Use the Central Limit Theorem to shows that we can approximate the co-CDF of PN i=1 Xi by the CDF of the standard normal distribution:

(b) (3 pts) Consider generating iid samples of Xi ∼ Poi(4). Show that it suffices to have N = 113 to ensure that PN i=1 Xi ≥ 400 holds true with a probability of at least 0.99. (Hint: Φ(2.326348) ≈ 0.99, and it suffices to carry out your numerical calculation to within 6 decimal places.)

Question 4 (14 pts). Let Xn ∈ Z≥0 be the amount of water (say measures in the unit of hundred million gallons) in MacRitchie Reservoir at noon on day n. During the 24 hours period beginning at this time, Yn ∈ Z≥0 units of water flow into the reservoir. The maximum capacity of the reservoir is 5 units, and excessive inflows are spilled and lost. Just before noon on each day, exactly 1 unit of water is removed if the reservoir is non-empty, and no water is removed if the reservoir is dried up Assume that Y1, Y2, . . . are iid discrete RVs, where1

Question 5. Consider transition probabilities P, Q on state-space S = {1, 2, 3, 4, 5, 6, 7, 8}

(a)Draw the auxiliary graph and identify the class(es). Is the DTMC irreducible?
(b)Which class(es) are closed? Hence or otherwise, list the recurrent and transient classes.

Question 6. Consider transition probabilities P, Q for irreducible DTMCS on state space S = {1, 2, 3, 4}:

For each of P, Q, answer the followings:
(a) (8 pts) Compute the invariant distributions for the DTMCs with transition probabilities P, Q.
(b) (4 pts) Does limn→∞ p (n) 1,2 exist? If yes, write down the limit. If not, explain which assumptions for the existence of the limit probability is/are violated.
(c) (4 pts) Does limn→∞ Pn i=1 1(Xi=1) n exist? If yes, write down the limit. If not, explain which assumptions for the existence of the limit probability is/are violated.