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Sma2110:Applied Mathematics Ii Question Paper
Sma2110:Applied Mathematics Ii
Course:Bachelor Of Computer Science
Institution: Meru University Of Science And Technology question papers
Exam Year:2012
QUESTION ONE (30 MARKS)
a) Explain what you understand by a stochastic process. (3 Marks) b) Let : 0 be a Poisson process with rate. Compute ( + ). (6 Marks) c) Let be the number of children born to the ith individual while denotes the population size at the Nth generation. Show that = + (2) by i. Using generating function techniques. (6 Marks) ii. Directly from the definition and the notion of conditional expectation. (7 Marks) d) For a Markov chain, show that ij nn P ixjxp 3 3 ? ??? . (8 Marks)
QUESTION TWO (20 MARKS)
a) Suppose that ; 0 and ; 0 are independent Poisson process with rates 1 and 2. Show that + ; = 0 is a Poisson process with rate 1 + 2. (10 Marks) b) Vehicles stopping by a roadsite restaurant in Meru form a Poisson process ; 0 with rate 1 = 20 per hour. A vehicle has 1, 2, 3, 4, or 5 persons on board with respective probabilities 0.3, 0.2, 0.3, 0.1 and 0.1. Compute the expected number of persons arriving at the restaurant within one hour. What can you advise the owner. (10 Marks)
2
QUESTION THREE (20 MARKS)
a) What is a Bernoulli process? (2 Marks) b) For any , , prove that knknn NprqNprkNpr ? ?? ? ?? 11 , where denotes the number of success in the first n trials of a Bernoulli Process. (12 Marks) c) Compute i. 53 (3 Marks) ii. 5,1123 (3 Marks)
QUESTION FOUR (20 MARKS)
a) Define the following terms i. A Markov process. (2 Marks) ii. A recurrent state (2 Marks) iii. A transient state (3 Marks) b) Prove the Chapman – Kolmogrov equation
kj
m
ik
n
ik
n ij P PP ? ? 1 (5 Marks) c) Let = ; be a Markov chain with state space = ,, and transition matrix
=
1 2
1 4
1 4
2 3
0
1 3
3 5
2 5
0
Compute 1 = ,2 = ,3 = ,5 = ,6 = ,7 = 0 = . (8 Marks)
QUESTION FIVE (20 MARKS)
a) Define what is meant by a probability generating function (pgf). (10 Marks) b) If X is a random variable distributed as binomial with parameters n and p. compute the pgf of X hence or otherwise compute i. () (3 Marks) ii. () (4 Marks) iii. 3 (3 Marks)
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