SAMPLE FIRST MIDTERM WITH SOLUTIONS
AMS 550
SPRING 2010
1. Two coins are tossed simultaneously and repeatedly. Coin i has a probability pi of showing
i
a head, 0 < pi < 1, i = 1, 2, Let Sn be the number of heads observed during the rst n tosses of
1 S 2 .
SAMPLE FIRST MIDTERM, AMS550
1. Two coins are tossed simultaneously and repeatedly. Coin i has a probability pi of showing
i
a head, 0 < pi < 1, i = 1, 2, Let Sn be the number of heads observed during the rst n tosses of
1 S 2 . Show that S is a Markov Ch
SAMPLE SECOND MIDTERM SOLUTIONS, AMS550, SPRING 2013
1. Buses and cars arrive to an intersection according to independent Poisson processes with
intensities and respectively. At time 0 the intersection is empty. With what probability will
the rst arrival
SECOND MIDTERM SOLUTIONS, AMS550
1. Let X be Bobs remaining lifetime without a Kidney and let Y be the time of availability of the Kidney. Then X and Y are independent, X exp(), and Y exp(). We have
P cfw_Y < X = + .
2. Given that N (7) = 5, the distribu
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Homework 2 Solution
4.18 If the state at time n is the nth coin to be flipped then sequence of consecutive states
constitute a two state Markov chain with transition probabilities:
P ,1 = .6 = 1 P , P2,1 = .5 = P2,2
1
1,2
(a) The stationary probabilities
Homework 1 (due 2/7)
1. Read sections 4.14.3 for the covered material and 4.44.7 for the future material.
2. Writing assignment:
Problems 3, 5, 6, 8, 13, 14, 15 from Chapter 4 (starting on page 275).
THIS HOMEWORK HAS BEEN UPLOADED TO BLACKBOARD. FUTURE
Homework 2 (due 2/14)
Read Sections 4.44.5 of the text.
Problems 18, 20, 22, 25, 31, 33 from Chapter 4.
Remarks:
Most of the problems are on the material of Section 4.4.
Proportion of time that an irreducible Markov chain spends in state i is i .
Homework 3 (due 2/21)
Read Problem 13 on p. 481 (Walds equation, Chapter 7) and read Sections 4.6 and 4.7 of
the text.
Writing assignment:
Problems 15 and 16 from Chapter 7 and 34, 45, 52 from Chapter 4.
Homework 4 Solution
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