A restaurant serves three fixed-price dinners costing c_1 = 12, c_2 = 15, and c_

Question

A restaurant serves three fixed-price dinners costing c_1 = 12, c_2 = 15, and c_3 = 20 dollars, respectively. For a randomly selected man/woman couple dining at this restaurant, let X = the cost of the man's dinner, and Y = the cost of the woman's dinner Suppose the joint probability mass function of X and Y is given by the matrix p = [0.05 0.05 0 0.05 0.1 0.2 0.1 0.35 0.1] The matrix element p_i, j is the probability that the man's dinner cost c_i and the woman's dinner cost c_j. What are the expected values X and Y and what is the covariance between X and Y? X = Y = Cov(X, Y) =

Explanation / Answer

Ans:

Marginal distribution of X=

P(X1)=0.05+0.05+0.1=0.2

P(X2)=0.05+0.1+0.35=0.5

P(X3)=0+0.2+0.1=0.3

Marginal distribution of Y

P(Y1)=0.05+0.05+0=0.1

P(Y2)=0.05+0.1+0.2=0.35

P(Y3)=0.1+0.35+0.1=0.55

Expected value of X=12*0.2+15*0.5+20*0.3=2.4+7.5+6=15.9

Expected value of Y=12*0.1+15*0.35+20*0.55=17.45

Cov(X,Y)=E[X,Y]-E[X]*E[Y]

E[X,Y]=12*12*0.05+12*15*0.05+12*20*0.1+15*12*0.05+15*15*0.1+15*20*0.35+20*12*0+20*15*0.2+20*20*0.1=276.7

Cov(X,Y)=276.7-15.9*17.45=-0.755

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