 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2021/paper-1/16/a/solution

Assume the three standard dice are fair and independent. For one die

X

1, \dots, 6

gives

E X = \frac{1 + 2 + 3 + 4 + 5 + 6}{6} = \frac{7}{2}, E X^{2} = \frac{1 + 4 + 9 + 16 + 25 + 36}{6} = \frac{91}{6} .

Var X = E X^{2} - (E X)^{2} = 35/12

. For the sum

Q = X_{1} + X_{2} + X_{3}

E Q = \frac{21}{2}, Var Q = 3 \frac{35}{12} = \frac{35}{4} .

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