Question

An instructor has given a short quiz consisting of two parts. For a randomly selected student, let X - the number of points earned on the first part and Y = the number of points earned on the second part. Suppose that the joint pmf of X and Y is given in the accompanying table. p(x, y) 0 5 10 15 0 0.03 0.06 0.02 0.10 x 5 0.04 0.17 0.20 0.10 10 0.01 0.15 0.11 0.01 (a) If the score recorded in the grade book is the total number of points earned on the two parts, what is the expected recorded score E(X+? (Enter your answer to one decimal place.) (b) If the maximum of the two scores is recorded, what is the expected recorded score? (Enter your answer to two decimal places.)

Answer 1

a) First we find the marginal distributions and Expectations of X and Y each

X	0	5	10
P(X)	0.21	0.51	0.28
x.P(X)	0	2.55	2.8

E(X) = ∑x.P(X) = 0 + 2.55 + 2.8
         = 5.35
E(X) = 5.35

Y	0	5	10	15
P(Y)	0.08	0.38	0.33	0.21
y.P(Y)	0	1.9	3.3	3.15

E(Y) = ∑y.P(Y) = 0 + 1.9 + 3.3 + 3.15  
         = 8.35  
E(Y) = 8.35  
  
E(X + Y) = E(X) + E(Y)  
          = 5.35 + 8.35  
          = 13.7  
E(X + Y) = 13.7  

b)    Let Z = max(X, Y)
We find the probability of Z by summing up the probabilities given for max value of X or Y      
e.g. Z= 5 => max (X, Y) = 5 thus we sum up probabilities for (X = 0, Y = 5), (X = 5, Y = 5), (X= 5, Y = 0)      
                                                                                         = 0.06 + 0.17 + 0.04

Z	0	5	10	15
P(Z)	0.03	0.27	0.49	0.21
Z.P(Z)	0	1.35	4.9	3.15

E(Z) = ∑z.P(Z) = 0 + 1.35 + 4.9 + 3.15
         = 9.4
E(Z) = 9.40