Find the solution for the Question below

Q. 7

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 thaI 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 binary data transmission system over a noisy channel using unipolar signals the signals are represented by p(t) = 0.5 V.0St SI, . q(t) = 0VOSIsI, The receiver uses a threshold device to detect binary values. Assuming zero mean additive white Gaussian noise with variance o? = 0.1, find a) the optimum threshold value b) the probability of a bit received being in error. (There is no matched filter in this problem)Problem 4: In a basic repetition code, every transmitted bit is repeated 3 times. that is: to transmit a III. the codeword i} is sent and to transmit a 1. the codeword 111 is sent The receiver takes the three received bits and decides which hit was sent via a majorityr vote of the three hits. Assume that the coded bits are transmitted at the rate of 3l'v'lbps and that for each bit, a transmission error may occur randomly and independently with probability of error p = l'g. [a]: Determine the following parameters of the code: - Bode rate; - Minimum distance; - Number of correctable errors; - Number of detects ble errors. {b} For n = 11.1.2.3. nd the probability of making to errors when transmitting a single 3-bit codeword. [c] Briefly explain the rationale behind the error correction mechanism (i.e.. the voting scheme]. For simplicity+ consider the transmission of a D input bit. {d} Find the probability that the receiver makes an incorrect decision? (10 points) 13. This question is on graphs and trees. What is the definition of a graph as given in Section 10.1? b) Provide an example of an application of a graph. c) Provide an example of an application of a tree. (See Section 1 1.1 of the text.) d) Graphs and trees have a plethora of applications in computer science and mathematics. As time permits, read Chapter 10 and 1 1. In particular, read about tree traversal. I encourage you to continue to read this material over the break. If you are studying computer science, this topic will reappear in your data structures and algorithm courses. There is nothing for you to turn in for this part.{4 points]4. A binary data transmission system over a noisy charmel using unipolar signals the signals are represented by M!) = 0513.0 at 51239) = DVJJ 5 rs}; The receiver uses a threshold device to detect binary values. Assuming zero mean additive white Gaussian noise with vaiiaace of = Ill , find a] the optimum threshold value b) the probability of a bit received being in error. {There is no matched lter in this problem]