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.
y | |||||
p(x, y) |
0 | 5 | 10 | 15 | |
x | 0 | 0.01 | 0.06 | 0.02 | 0.10 |
5 | 0.04 | 0.14 | 0.20 | 0.10 | |
10 | 0.01 | 0.15 | 0.16 | 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 + 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.)
a) The value of X + Y is computed for each combination of X, Y here. The expected value for X + Y then is computed here as:
p(x,y) | 0 | 5 | 10 | 15 |
0 | 0.01 | 0.06 | 0.02 | 0.1 |
5 | 0.04 | 0.14 | 0.2 | 0.1 |
10 | 0.01 | 0.15 | 0.16 | 0.01 |
x + y | 0 | 5 | 10 | 15 |
0 | 0 | 5 | 10 | 15 |
5 | 5 | 10 | 15 | 20 |
10 | 10 | 15 | 20 | 25 |
E(X + Y) | 0 | 5 | 10 | 15 |
0 | 0 | 0.3 | 0.2 | 1.5 |
5 | 0.2 | 1.4 | 3 | 2 |
10 | 0.1 | 2.25 | 3.2 | 0.25 |
We add the values in the last table here to get the expected value as:
Therefore 14.4 is the required expected value here.
b) For the maximum of the two scores, we calculate the expected score here as:
Max(X, Y) | 0 | 5 | 10 | 15 |
0 | 0 | 5 | 10 | 15 |
5 | 5 | 5 | 10 | 15 |
10 | 10 | 10 | 10 | 15 |
E(max(X, Y)) | 0 | 5 | 10 | 15 |
0 | 0 | 0.3 | 0.2 | 1.5 |
5 | 0.2 | 0.7 | 2 | 1.5 |
10 | 0.1 | 1.5 | 1.6 | 0.15 |
Therefore 9.75 is the required expected value here.
An instructor has given a short quiz consisting of two parts. For a randomly selected student,...
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. y p(x, y) 0 5 10 15 x 0 0.01 0.06 0.02 0.10 5 0.04 0.14 0.20 0.10 10 0.01 0.15 0.16 0.01 (a)...
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. y p(x,y) 0 5 10 15 0 0.01 0.06 0.02 0.10 х 5 0.04 0.17 0.20 0.10 10 0.01 0.15 0.13 0.01 (a) If the...
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. y p(x,y) 0 5 10 15 0 0.03 0.06 0.02 0.10 х 5 0.04 0.13 0.20 0.10 10 0.01 0.15 0.15 0.01 (a) If the score recorded...
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 у 0 5 10 0.03 0.06 0.02 0.04 0.13 0.20 0.01 0.15 0.15 15 0.10 Х 5 0.10 0.01 10 (a) If...
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