Simple linear regression is a statistical method that allows us to summarize and study relationships between two continuous (quantitative) variables: One variable, denoted X, is regarded as the predictor, explanatory, or independent variable. The other variable, denoted Y, is regarded as the response, outcome, or dependent variable. Suppose that we are given n-i.i.d observations { (x;, y;)}"_, from the assumed simple linear regression model Y = BIX + Bo + . Answer the following questions on simple linear regression. 5-a. Denote 1 and Bo as the point estimators of B, and Bo, respectively, that are obtained through the least squares method. Show, step by step, that the two point estimators are unbiased. Derive the least squares estimator of of and determine whether it is unbiased or not. Show your work step by step. 5-b. Calculate _'_1(yi - Bix; - Bo) (Bix, + Bo). Determine whether the point (X, Y) is on the line Y = 1X + Bo. Explain your reasoning mathematically. 5-c. Using the maximum likelihood estimation (MLE) technique, derive a point estimator for the coefficient B1 and the intercept Bo, respectively. Determine whether the point estimators that you obtained via MLE are unbiased or not. Justify your conclusion mathematically. 5-d. Calculate the variance of the four estimators from Questions 5-a and 5-c, respectively. Show your work step by step. 5-e. Suppose that we are using the simple linear regression model Y = B1 X + Bo + 1 while the true model is Y = 1X1 + B2X2 + Bo + 82 where Bo, B1, and B2 are constants. We assume that the distributions of &, and e2 are both N(0,02), i.e., normal distribution with variance o?. We further assume that the two noise variables are uncorrelated. Find the least squares estimator of B, in this case and determine whether the point estimator that you obtain is biased or not. If it is biased, calculate the bias.A restaurant faces very high demand for its signature mousse desserts in the evening but is less busy during the day. Its manager estimates that inverse demand functions are pe = 30 - Qe in the evening and pd = 16 -Qd during the day, where e and d denote evening and daytime. The marginal cost of producing its dessert evening, MCe, is $8. The marginal cost of producing its dessert daytime, MCd, is $4. There is no fixed cost of producing dessert. Create a spreadsheet with the column headings Qe, Pe, TRe, MRe, TCe, MCe, ne, Qd, Pd, TRd, MRd, TCd, MCd, and nd. (note: ne is profit evening and nd indicates profit daytime) a. What are the optimal prices for the dessert that the restaurant should charge during the evening hours? b. What is the optimal quantity for the dessert that the restaurant should produce during the evening hours? c. What is the total cost of producing the optimal quantity for the dessert during the evening hours? d. What is the maximum profit for the dessert that the restaurant should produce during the evening hours? e. What are the optimal prices for the dessert that the restaurant should charge during the daytime hours? f. What is the optimal quantity for the dessert that the restaurant should produce during the daytime hours? I g. What is the total cost of producing the optimal quantity for the dessert during the daytime hours? h. What is the maximum profit for the dessert that the restaurant should produce during the daytime hours?4. (30 points) Consider the following game. There are ten dollars to divide. Two players are each required to simultaneously name an integer between 0 and 10. The player who names the higher number gets to keep the money. If they name the same number, the money is equally shared between them. (a) Describe the set of players N, the set of strategies { Silien, and the payoff function QuitiEN. (b) Are there strategies that are strictly dominated? Demonstrate your reasoning. What are the resulting strategies after iterated elimination of strictly dominated strategies? (c) Find the best responses (correspondence) for each player. That is, find the strategies that maximize a player's payoff given what the other player does. (d) Find the Nash equilibria of the game. (e) Suppose now the game is changed. Whenever there is a tie, each player receives nothing. Answer the same questions in parts (b) and (c). Find the pure-strategy Nash equilibria of the game