1. Let Xi = lunch condition on day i (rice/noodles)

P(Xi+1 = rice| Xi-1 = rice, Xi = rice) = 0.7

P(Xi+1 = rice| Xi-1 = noodles, Xi = rice) = 0.6

P(Xi+1 = rice| Xi-1 = rice, Xi = noodles) = 0.3

P(Xi+1 = rice| Xi-1 = noodles, Xi = noodles) = 0.55

Q1. Is {Xn} a Markov Chains? Why?

Q2. How to transform the process into a M.C. ?

a) What is the probability that a hand of 13 cards contains four of a kind (e.g., four 5’s, four Kings, four aces, etc.)?

b) A single card is randomly drawn from a thoroughly shuffled deck of 52 cards. What is the probability that the drawn card will be either a diamond or a queen?

c) The probability that the events A and B both occur is 0.3. The individual probabilities of the events A and B are 0.7 and 0.5. What is the probability that neither event A nor event B occurs?

1.Plotting densitiesPlot the probability mass function (pmf) or probability density function (pdf) for eachof the following scenarios:(a) Consider abinomialrandom variable,X.i. Plot the pmf ofX∼Bin(n= 10,p= 0.3).ii. Plot the pmf ofX∼Bin(n= 10,p= 0.7).iii. Plot the pmf ofX∼Bin(n= 100,p= 0.3).iv. What happens to the shape of the pmf ofX∼Bin(n,p) whenpgets larger?v. What happens whenngets larger?(b) Consider ageometricrandom variable,Y.i. Plot the pmf ofY∼Geom(p= 0.1).ii. Plot the pmf ofY∼Geom(p= 0.5).iii. Plot the pmf ofY∼Geom(p= 0.8).iv. What happens to the shape of the pmf ofY∼Geom(p) whenpgets larger?(c) Consider aexponentialrandom variable,T.i. Plot the pdf ofT∼Exp(λ= 0.1).ii. Plot the pdf ofT∼Exp(λ= 0.5).iii. Plot the pdf ofT∼Exp(λ= 2).iv. What happens to the shape of the pdf ofT∼Exp(λ) whenλgets larger?(d) Consider anormalrandom variable,M.i. Plot the pdf ofM∼N(μ= 2,σ2= 1).ii. Plot the pdf ofM∼N(μ=−1,σ2= 1).iii. Plot the pdf ofM∼N(μ= 2,σ2= 5).1 iv. What happens to the pdf ofM∼N(μ,σ2) whenμis changed?v. What happens to the pdf ofM∼N(μ,σ2) whenσ2gets larger?(e) Which of the continuous distributions looks the most similar to the geometricdistribution? Which looks the most similar to the binomial distribution (withlargen)? Do these relationships make sense, based on your knowledge of thedistributions and their assumptions?

Melissa recently paid $625 for round-trip airfare to San Francisco to attend a business conference for three days. Melissa also paid the following expenses: $725 fee to register for the conference, $415 per night for three nights’ lodging, $205 for meals, and $395 for cab fare. (Leave no answers blank. Enter zero if applicable.) (Round your answer to the nearest dollar amount.)

a. What amount of the travel costs can Melissa deduct as business expenses?

b. Suppose that while Melissa was on the coast, she also spent two days sightseeing the national parks in the area. To do the sightseeing, she paid $1,590 for transportation, $880 for lodging, and $390 for meals during this part of her trip, which she considers personal in nature. What amount of the travel costs can Melissa deduct as business expenses?

c. Suppose that Melissa made the trip to San Francisco primarily to visit the national parks and only attended the business conference as an incidental benefit of being present on the coast at that time. What amount of the airfare can Melissa deduct as a business expense?

d. Suppose that Melissa’s permanent residence and business was located in San Francisco. She attended the conference in San Francisco and paid $725 for the registration fee. She drove 105 miles over the course of three days and paid $169 for parking at the conference hotel. In addition, she spent $295 for breakfast and dinner over the three days of the conference. She bought breakfast on the way to the conference hotel and she bought dinner on her way home each night from the conference. What amount of the travel costs can Melissa deduct as business expenses? (Use standard mileage rate. Do not round intermediate calculations. Round your final answer to the nearest dollar amount.)

Melissa recently paid $625 for round-trip airfare to San Francisco to attend a business conference for three days. Melissa also paid the following expenses: $405 fee to register for the conference, $360 per night for three nights’ lodging, $280 for meals, and $240 for cab fare. (Leave no answers blank. Enter zero if applicable.) (Do not round intermediate calculations. Round your final answer to the nearest dollar amount.)

a. What amount of the travel costs can Melissa deduct as business expenses?
b. Suppose that while Melissa was on the coast, she also spent two days sightseeing the national parks in the area. To do the sightseeing, she paid $1,650 for transportation, $1,035 for lodging, and $320 for meals during this part of her trip, which she considers personal in nature. What amount of the travel costs can Melissa deduct as business expenses?
c. Suppose that Melissa made the trip to San Francisco primarily to visit the national parks and only attended the business conference as an incidental benefit of being present on the coast at that time. What amount of the airfare can Melissa deduct as a business expense?
d. Suppose that Melissa’s permanent residence and business was located in San Francisco. She attended the conference in San Francisco and paid $405 for the registration fee. She drove 117 miles over the course of three days and paid $152 for parking at the conference hotel. In addition, she spent $625 for breakfast and dinner over the three days of the conference. She bought breakfast on the way to the conference hotel and she bought dinner on her way home each night from the conference. What amount of the travel costs can Melissa deduct as business expenses? (Use standard mileage rate.)

Melissa recently paid $760 for round-trip airfare to San Francisco to attend a business conference for three days. Melissa also paid the following expenses: $370 fee to register for the conference, $365 per night for three nights’ lodging, $250 for meals, and $300 for cab fare. (Leave no answers blank. Enter zero if applicable.) (Do not round intermediate calculations. Round your final answer to the nearest dollar amount.)

A. What amount of the travel costs can Melissa deduct as business expenses?
Deductible Amount:

B. Suppose that while Melissa was on the coast, she also spent two days sightseeing the national parks in the area. To do the sightseeing, she paid $1,710 for transportation, $1,285 for lodging, and $475 for meals during this part of her trip, which she considers personal in nature. What amount of the travel costs can Melissa deduct as business expenses?
Deductible Amount:

C. Suppose that Melissa made the trip to San Francisco primarily to visit the national parks and only attended the business conference as an incidental benefit of being present on the coast at that time. What amount of the airfare can Melissa deduct as a business expense?
Deductible Amount:

D. Suppose that Melissa’s permanent residence and business was located in San Francisco. She attended the conference in San Francisco and paid $370 for the registration fee. She drove 153 miles over the course of three days and paid $190 for parking at the conference hotel. In addition, she spent $430 for breakfast and dinner over the three days of the conference. She bought breakfast on the way to the conference hotel and she bought dinner on her way home each night from the conference. What amount of these costs can Melissa deduct as business expenses? (Use standard mileage rate.)
Deductible Amount:

In the flue season, people with vaccine protection have probability of 0.2 to be infected by the flue virus, whereas people without vaccine protection have probability of 0.6 to be infected by the virus. For a given population, the proportion of people being vaccinated is 0.7.

a.) Given that a person is infected by the flue virus, what is the probability that the person has not taken the vaccine?

b.) Randomly select one person from this population, what is the probability that this person is infected by the flue virus?

This section has to do with spherical mirrors and lenses. So here goes...

(a) In front of a spherical concave mirror of radius 23 cm, you position an object of height 2.8 cm somewhere along the principal axis. The resultant image has a height of 2.2 cm. How far from the mirror is the object located? Answer: ____________.__ cm

(b) In front of a spherical convex mirror of radius 23 cm, you position an object of height 2.8 cm somewhere along the principal axis. The resultant image has a height of 2.2 cm. How far from the mirror is the object located? (10 points) Answer: ____________.__ cm

(c) On one side of a converging lens of focal length 12 cm, you position an object of height 0.7 cm somewhere along the principal axis. The resultant image has a height of 0.2 cm. How far from the lens is the object located? Answer: ____________.__ cm

(d) On one side of a diverging lens of focal length 12 cm, you position an object of height 0.7 cm somewhere along the principal axis. The resultant image has a height of 0.2 cm. How far from the lens is the object located? Answer: ____________.__ cm

We wish to study the effect that the size of various U.S. cities have over ozone concentrations. The size of the populations correspond to the last census (in millions of inhabitants), and the average quantity of ozone per hour was determined in (parts per thousand millions);

a) Identify the variables X and Y and analyze if the variables are linearly related.

b) Determine the line of regression and interpret the coefficients of the line.

c) Estimate the average quantity of ozone per hour in a population of 1.000.000 inhabitants.

Mr. John Backster, a retired executive, desires to invest a portion of his assets in rental property. He has narrowed his choices to two apartment complexes, Windy Acres and Hillcrest Apartments. The anticipated annual cash inflows from each are as follows:

Mr. is likely to hold the apartment complex of his choice for about 25 years and will use this period for decision making purposes. Either apartment can be purchased for $140,000. Mr. Backster uses risk-adjusted discount rate approach when evaluating investments. His scale is related to the coefficient of variation ( for other types of investments, he also considers other measures).

a) compute the risk-adjusted net present value for Windy Acres and Hillcrest Apartments.

b) Which investment should Mr. Backster accept if the two investments are mutually exclusive? If the investments are not mutually exclusive and no capital rationing is involved, how would your decision be affected?