Coastal report solutions. The weather at a coastal resort is classified each day simply as "sunny" or "rainy." A

sunny day is followed by another sunny day with probability 0.9, and a rainy day is

followed by another rainy day with probability 0.3. (a) Describe this as a Markov chain.

(b) If Friday is sunny, what is the probability that Sunday is also sunny? (c) If Friday is

sunny, what is the probability that both Saturday and Sunday are sunny?

2 At another resort, it is known that the probability that any two consecutive days are both

sunny is 0.7 and that the other three combinations are equally likely. Find the transition

probabilities.

3 A machine produces electronic components that may come out defective and the process

is such that defective components tend to come in clusters. A defective component is

followed by another defective component with probability 0.3, whereas a nondefective

component is followed by a defective component with probability 0.01. Describe this

as a Markov chain, and find the long-term proportion of defective components.

4 An insurance company classifies its auto insurance policyholders in the categories

"high," "intermediate," or "low" risk. In any given year, a policyholder has no accidents

with probability 0.6, one accident with probability 0.2, two accidents with probability

0.1, and more than two accidents with probability 0.1. If you have no accidents, you

are moved down one risk category; if you have one, you stay where you are; if you

have two accidents, you move up one category; and if you have more than two, you

always move to high risk. (a) Describe the sequence of moves between categories of

a policyholder as a Markov chain. (b) If you start as a low-risk customer, how many

years can you expect to stay there? (c) How many years pass on average between two

consecutive visits in the high-risk category?

5 Consider the ON/OFF system from Example 8.2.4. Let Xn be the state after n steps,

and define Yn = (Xn, Xn+1). Show that {Yn} is a Markov chain on the state space

{0, 1} {0, 1}, find its transition matrix and stationary distribution.

6 Suppose that state i is transient and that i ? j. Can j be recurrent?

7 Consider the state space S = {1, 2, ..., n}. Describe a Markov chain on S that has only

one recurrent state.

(a) (4 Points) Explain why when the input to an LTI system is a White Gaussian Random Process, the output is also a Gaussian Random Process. Is the output also White? Why? (b) (4 Points) Show that the first statement in Part (a) is true for any (not necessarily White) Gaussian Random Process.6.5 Comparison of Gaussian random variables 5 points Let X be a Gaussian random variable with my = 10 and ox = 3. Instead, let Y be a Gaussian random variable with my = 8 and ox = 6. a) Find all values a such that fx (o) = fy(a). b) Find all values 8 such that P(X