MSIN0022 - Mathematics III (Probability Theory)
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MSIN0022: Mathematics III (Probability Theory)
Examination Paper
2020/21
1. [20 marks] We let {Tn : n > 1} be a sequence of independent exponential random variables, each having mean E[Tn] = 1. We define for n > 4, the random variable Sn = 1 if Tn = max{Tn, Tn − 1 , Tn −2 , Tn −3 }; otherwise, we define Sn = 0. Also let S1 = S2 = S3 = 0.
(a) [10 marks] Determine the mean and variance of Sn for n > 4.
(b) [10 marks] Determine the covariance Cov(Sn, Sn+1) for n > 4.
2. [15 marks] We consider a sequence of random variables X1, X2 , . . . consisting of independent exponential random variables with rate 1. We let m > 0, and define the random variable M by
M = min{n : Xn > m}.
(a) [5 marks] What is the PMF of M?
(b) [5 marks] What is the PDF of XM?
(c) [5 marks] Is M independent of XM?
3. [15 marks] Let X and Y be jointly continuous with joint PDF:
f(x, y) =
(a) [5 marks] What is the value of c?
(b) [5 marks] What is P(X + Y > 1)?
(c) [5 marks] Are X and Y independent?
4. [15 marks] Let X1 and X2 be two independent exponentially distributed random variables, with means E[X1] = 1 and E[X2] = 1/2. Let Y = min{X1, X2 } and Z = max{X1, X2 }.
(a) [5 marks] What is E[Y]?
(b) [5 marks] What is E[Z]?
(c) [5 marks] What is the cdf of Y?
5. [20 marks] In this problem, we study the evolution of weather states. To simplify, we assume that there are only two possible states of weather: Either rainy or dry. Moreover, suppose that the state of the weather on a given day is influenced by the weather states on the previous two days. We let R denote the rainy state, and D denote the dry state, and Xn be the weather on day n. The weather evolves as follows:
● P(Xn+1 = RlXn = R, Xn − 1 = R) = 0.7
● P(Xn+1 = RlXn = R, Xn − 1 = D) = 0.5
● P(Xn+1 = RlXn = D, Xn − 1 = R) = 0.4
● P(Xn+1 = RlXn = D, Xn − 1 = D) = 0.2
(a) [10 marks] Calculate the probability P(Xn+1 = R, Xn+2 = D, Xn+3 = DlXn − 1 =
R, Xn = R).
(b) [10 marks] Explain why {Xn : n > 0} is not a Markov chain. Construct a Markov
chain, based on the information above, by re-defining the state of the Markov chain (instead of the state Xn e {R, D} above). Specify the probability transition matrix of this new chain.
6. [15 marks] Consider X to be a uniformly distributed random variable over (0 , 1). Define the random variable Y = - ln(X). Can you find the PDF of Y?
2022-04-20