The total cost of building a house (T) is the cost of materials (X) plus the cost of labor (Y) plus fixed costs (such as the costs of permits). X and Y are random variables. Fixed costs are not random. The joint probability distribution of X and Y is given below. The covariance between X and Y is -0.7 (a negative covariance). You'll need this covariance below so remember to come back and get it! Y 1 2 3 4 2 0 0 0.1 0.2 X 3 01 0.1 0.1 0.1 4 0.2 0.1 0 0 1-1 Give the marginal probability distribution of X. Give the marginal probability distribution of Y. (Either state these distributions or copy the joint probability distribution of X and Y onto your paper and add the marginal probability distributions. Be sure to label them.) (6 points) 1-2 Calculate the expected cost of materials (expected value of X). (3 points) 1-3 Calculate the variance of the cost of materials (variance of X). (3 points) 1-4 Give the conditional probability distribution of X. given Y equals 2. (3 points) 1-5 Calculate the expected cost of materials given that the cost of labor is 2. (3 points) 1-6 Calculate the variance of the costs of materials given that the cost of labor is 2. (3 points) 1-7 Are X and Y independent? Explain (demonstrate) why or why not. (4 points) 1-8 Fixed costs are 2. The total cost of building a house (T) is the sum of materials costs (X), labor costs (Y), and fixed costs. Calculate expected total costs (expected value of T). (5 points) 1-9 Calculate the variance of total costs (variance of T). (5 points) 1-10 How many possible values of T have a positive probability of occurring? Do not count a value (realization) of T if the probability of that value is zero. What is the probability that total costs are less than 8? (6 points) 1-11 The joint probability distribution of X and Y and the value of fixed costs represent a contract that the builder offers you. Suppose the builder offers you a different contract, given by the new probability distribution below. (Fixed costs remain the same.) Calculate the covariance between X and Y. (6 points) 1-11 The joint probability distribution of X and Y and the value of fixed costs represent a contract that the builder offers you. Suppose the builder offers you a different contract, given by the new probability distribution below. (Fixed costs remain the same.) Calculate the covariance between X and Y. (6 points) 1 Y 2 3 4 1 0 0 01 0.2 X 3 do 0.1 0.1 0.1 0.1 5 0.2 0.1 0 0 1-12 You decide to accept the new contract if it increases the probability that total costs are less than 8. What is the probability that total costs are less than 8 with the new contract? (3 points) The total cost of building a house (T) is the cost of materials (X) plus the cost of labor (Y) plus fixed costs (such as the costs of permits). X and Y are random variables. Fixed costs are not random. The joint probability distribution of X and Y is given below. The covariance between X and Y is -0.7 (a negative covariance). You'll need this covariance below so remember to come back and get it! Y 1 2 3 4 2 0 0 0.1 0.2 X 3 01 0.1 0.1 0.1 4 0.2 0.1 0 0 1-1 Give the marginal probability distribution of X. Give the marginal probability distribution of Y. (Either state these distributions or copy the joint probability distribution of X and Y onto your paper and add the marginal probability distributions. Be sure to label them.) (6 points) 1-2 Calculate the expected cost of materials (expected value of X). (3 points) 1-3 Calculate the variance of the cost of materials (variance of X). (3 points) 1-4 Give the conditional probability distribution of X. given Y equals 2. (3 points) 1-5 Calculate the expected cost of materials given that the cost of labor is 2. (3 points) 1-6 Calculate the variance of the costs of materials given that the cost of labor is 2. (3 points) 1-7 Are X and Y independent? Explain (demonstrate) why or why not. (4 points) 1-8 Fixed costs are 2. The total cost of building a house (T) is the sum of materials costs (X), labor costs (Y), and fixed costs. Calculate expected total costs (expected value of T). (5 points) 1-9 Calculate the variance of total costs (variance of T). (5 points) 1-10 How many possible values of T have a positive probability of occurring? Do not count a value (realization) of T if the probability of that value is zero. What is the probability that total costs are less than 8? (6 points) 1-11 The joint probability distribution of X and Y and the value of fixed costs represent a contract that the builder offers you. Suppose the builder offers you a different contract, given by the new probability distribution below. (Fixed costs remain the same.) Calculate the covariance between X and Y. (6 points) 1-11 The joint probability distribution of X and Y and the value of fixed costs represent a contract that the builder offers you. Suppose the builder offers you a different contract, given by the new probability distribution below. (Fixed costs remain the same.) Calculate the covariance between X and Y. (6 points) 1 Y 2 3 4 1 0 0 01 0.2 X 3 do 0.1 0.1 0.1 0.1 5 0.2 0.1 0 0 1-12 You decide to accept the new contract if it increases the probability that total costs are less than 8. What is the probability that total costs are less than 8 with the new contract? (3 points)