The LSU Tigers baseball team will play the Alabama baseball team in a weekend series of two games. Let \(W=0,1\), or 2 equal the number of games LSU wins. Let the weekend's weather be designated as Cold or Not Cold. Let \(C=1\) if the weather is cold and \(C=0\) if the weather is not cold. The joint probability function of these two random variables is given in Table P.9, along with space for the marginal distributions.

a. Fill in the blanks, (i)-(vi).

b. Using the results of (a), find the conditional probability distribution of the number of wins, \(W\), conditional on the weather being warm, \(C=0\). Based on a comparison of the conditional probability distribution \(f(w \mid C=0)\) and the marginal distribution \(f(w)\), can you conclude that the number of games LSU wins \(W\) is statistically independent of the weather conditions \(C\), or not? Explain.

c. Find the expected value of the number of LSU wins, \(W\). Also find the conditional expectation \(E(W \mid C=0)\). Show your work. What kind of weather is more favorable for the LSU Tigers baseball team?

d. The revenue of vendors at the LSU Alex Box baseball stadium depends on the crowds, which in turn depends on the weather. Suppose that food sales \(F O O D=\$ 10,000-3000 C\). Use the rules for expected value and variance to find the expected value and standard deviation of food sales.

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TABLE P.9 Table for Exercise P.13 W = 0 W = 1 W = 2 f(c) C = 1 (i) 0.12 0.12 (ii) C=0 0.07 0.14 (iii) (iv) f(w) (v) (vi) 0.61