 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2001/paper-1/12/b/solution

The triangular probability density function is normalized because

2 \int_{0}^{1} (1 - x) d x = 1

. It is even, so the mean integrand is odd and

μ = 0

. Its variance is consequently its second moment:

μ = 0, σ^{2} = 2 \int_{0}^{1} x^{2} (1 - x) d x = 2 (\frac{1}{3} - \frac{1}{4}) = \frac{1}{6} .

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