A company sells a number of low-cost, high volume cell phone accessory products.For one such product, it is equally likely that the annual unit sales will be low or high.If sales are low (60,000), the company can sell the product for $10 per unit.If sales are high (100,000), a competitor will enter the market and the company will be able to sell the product for only $8 per unit.

The variable cost per unit has a 25% chance of being $6, a 50% change of being $7.50, and a 25% chance of being $9.Annual fixed costs are $30,000.

Develop the Simulation model to estimate the company's expected annual profit. Using the @RISK software, run 1000 iterations, 1 simulation.Label the worksheet "Model"

Run Browse Results (under EXPLORE in @RISK menu) and save the graph and grid to a new worksheet. Label the worksheet "Browse Results"

Run Results Summary (under EXPLORE in @RISK menu) and save the output to a worksheet in the active workbook.

Find the 95% confidence interval for the company's annual profit. Clearly label your answer on the "Model" worksheet in a text box.

Suppose that annual unit sales, variable cost and unit price are equal to their respective expected values - that is, you assume no uncertainty. Determine the company's annual profit for this scenario.Enter the information used to calculate your answer on the "Model" worksheet and clearly label your answer.

Can you conclude from the results in parts 1 and 5 that the expected profit from a simulation is equal to the profit from a scenario where each input assumes it's expected value?Clearly label your answer with your explanation in a text box on the "Model" worksheet

Suppose you observe an i.i.d. sample X1, ..., Xn of a Bernoulli random variable X with unknown parameter 40 = P (X = 1). We use the Bayesian approach to statistical inference and model the unknown parameter do as a random variable @ (defined jointly with X). That is, let the conditional distribution of X given O be X|0 ~ Ber (@) and let TO (0) = 30 1 (0 E [0, 1]) be the prior distribution of O. Let P denote the probabilities implied by the Bayesian model. Useful Facts: The Beta (a, ) distribution has pdf f ( t ) = - B(a, B) where B(a, B) = 1 1 (1 - 2) . at, and Mean a+B aB Variance = (a + B) (a + 8+ 1) Mode = argmaxf= a-1 - for a, B > 1. a+8-2 The mode is the value of the random variable at which the pdf attains its maximum. Posterior Distribution The posterior distribution Tex,,...,x. of @ given the sample X1, . .., Xn is a Beta distribution Beta (a, b). Specify the parameters a and b below. (Enter barX_n for Xn.) b = Bayes Estimator and its Limit What is the mean On Bayes of the posterior distribution in terms of Xn? (Enter barX_n for X,.). = ["xle H = sakeg The estimator , converges in probability to a constant as n -> oo? What is this limiting constant? (Enter in terms of the true parameter Op.) (7) in P1. (10 marks) Suppose that you have been given the following Bayesian network, containing four vari- ables. Three of the variables are Boolean (true or false), but variable C has three possible values: Chigh, Cmed and Chow. The information from the conditional probability tables is the following: . P(a) = 0.6 . P(b | a) = 0.7, P(b | -a) = 0.2 . P(chigh | a) = 0.3, P(Cmed | a) = 0.5, P(clow | a) = 0.2, P(Chigh | 70) = 0.1, P(Cmed | 70) = 0.2, P(Clow | 70) = 0.7 . P(d | b, Chigh) = 0.2, P(d | b, Cmed) = 0.4, P(d | b, Clow) = 0.8, P(d | -b, Chigh) = 0.1, P(d | -b, (med) = 0.6, P(d | -b, Clow) = 0.9 Use the method of exact inference by enumeration discussed in class to determine: P(chigh | -d), P(cmed | ~d), and P(Glow | -d). You must show your work.Problem 5. Suppose that you are given a Bayesian network for Boolean random variables that corresponds to the following factorization: P(A, B, C, D) = P(A) P(B | A) P(C | A) P(D | B, C) | Assume that the conditional probability tables (CPTs) for the Bayes net are as follows: P(A) AB P(B A) AC P(C A) BCD P(D B, C) 0.1 0 0 0.2 0 0 0.4 000 0.9 0.9 01 0.8 0 1 0.6 001 0.1 10 0.7 10 0.5 010 0.8 1 1 0.3 1 1 0.5 01 1 0.2 100 0.7 101 0.3 1 10 0.6 1 1 1 0.4 Use exact inferencing to answer the following. a) Solve for the probability value P(4=1, B=0, C=1, D=0). b) Solve for the probability value P(4=1, B=0, D=0). c) Solve for the probability distribution P(B). d) Solve for the conditional probability distribution P(B | D=1)