An instructor has given a short quiz consisting of two parts

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. (a) If the score recorded in the grade book is the total number or points earned on the two parts, what is the expected recorded score E(X + Y)? (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.)

Solution

the maximum is 0 when x=y=0 with P=0.01 the maximum is 5 when x y 0 5 5 0 5 5 with P=0.06+0.04+0.13=0.23 the maximum is 10 when x y 0 10 5 10 10 10 10 0 10 5 with probability P=0.02+0.20+0.01+0.15+0.17=0.22+0.33=0.55 the maximum is 15 when x y 0 15 5 15 10 15 with probability 0.1+.1+0.01=0.21 So the distribution of the maximum is max 0 5 10 15 P 0.01 0.23 0.55 0.21 (check indeed the sum of probabilities is 1) So the expectation is 0*0.01+5*0.23+10*0.55+15*0.21=9.8