Avoid rounding errors between steps. Round your final answers to 4 decimal places as needed. 4.1. A random variable X has the following probability distribution. x P(X = x) 2 0.2 4 0.2 6 0.3 8 0.3 4.1.a Compute the expected value of X. 4.1.b (0.3 pt) Compute the variance of X. 4.1.c Compute the standard deviation of X. . 4.2. True or False. Stock A's and Stock B's average returns are both 5%. Stock A's return has a higher standard deviation than Stock B's return. If a decision maker decides to choose a stock with a more stable return, then he will choose Stock A. . 4.3. A video rental store has two video cameras available for customers to rent. Historically, demand for cameras has followed the distribution shown in the table. The revenue per rental is $40. If a customer wants a camera and none is available, the store gives a $15 coupon for snacks. Demand Relative Frequency Revenue Cost 0 0.35 0 0 1 0.30 40 0 2 0.20 80 0 3 0.10 80 15 4 0.05 80 30 What is the expected profit from camera rentals? (Hint: profit = revenue cost. Calculate the profit for each demand scenario before finding the expected profit.) 4.4. ABC Inc. must make a decision on its current capacity for next year. ABC is consider ing three alternatives. For each alternative, estimated profits (in $000s) based on next year's demand are shown in the table below. ABC Inc. estimates that the probability of low demand is 0.4 and the probability of high demand is 0.6. Next Year's Demand . Alternative . Low . High . A1:Expand . $97 . $192 . A2:Subcontract . $125 . $175 . A3: Do Nothing . $120 . $147 . 4.4.a True of False. Alternative 2 (A2) dominates Alternative 3 (A3). . 4.4.b Which alternative should be chosen based on the optimistic (maximax) decision criterion? . 4.4.c Which alternative should be chosen based on the conservative (maximin) decision criterion? . 4.4.d Construct the regret table. What is the maximum regret if ABC chooses A1. [Give a numerical answer in $.] . 4.4.e Which alternative should be chosen based on the minimax regret crite rion? . 4.4.f Which alternative should be chosen using the expected monetary value criterion? 4.5. Steves Mountain Bicycle Shop is considering three options for its facility next year. Steve can expand his current shop, move to a larger facility, or make no change. With a good market, the annual payoff would be $76,000 if he expands, $90,000 if he moves, and $40,000 if he does nothing. With an average market, his payoffs will be $30,000, $41,000, and $15,000, respectively. With a poor market, his payoff will be $17,000, $28,000, and $2,000. . 4.5.a What is the best option based on optimistic criterion? . 4.5.b What is the best option based on conservative criterion? . 4.5.c What is the best option using minimax regret criterion? . 4.5.d Suppose the probability of a good market is 0.4 and the probability of an average market is 0.35. What is the best option under the expected value criterion