An Instructor has given a short quiz consisting of two parts. For a randomly sel

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. (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 + 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.)

Explanation / Answer

Solution:

a) For E(X+Y) we have calculate the following thing

For E(X)

P(X=0)= sum of probabilities of zero= 0.02+0.06+0.02+0.10=0.2

P(X=5)= sum of probabilities of five= 0.04+0.14+0.20+0.10=0.48

P(X=10)= sum of probabilities of ten= 0.01+0.15+0.15+0.01= 0.32

Now E(X)= (0*0.2)+(5*0.48)+(10*.32) =5.6

For E(Y)

P(X=0)= sum of probabilities of zero= 0.02+0.04+0.01= 0.07

P(X=5)= sum of probabilities of five= 0.06+0.14+0.15= 0.35

P(X=10)= sum of probabilities of ten= 0.02+0.20+0.15= 0.37

P(X=15)= sum of probabilities of fifteen= 0.10+0.10+0.01 = 0.21

E(Y)= (0*0.07)+(5*0.35)+(10*0.37)+(15*.21) =0+1.75+3.7+3.15= 8.6

E(X+Y)= E(X)+E(Y)

=5.6+8.6

E(X+Y) =14.2

This is the expected recorded score.

b) Maximum off the two score recorded here for X=10 and Y=15

therefor P(X=10, Y=15) =0.01

This is the maximum recorded score.

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