 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/13/b/i/solution

Retain the original PDF's full-line integration limits. For

n \geq 2

, write

x^{n} e^{- x^{2}} d x = - x^{n - 1} d (e^{- x^{2}}) /2

I_{n} = [- \frac{1}{2} x^{n - 1} e^{- x^{2}}]_{- \infty}^{\infty} + \frac{n - 1}{2} \int_{- \infty}^{\infty} x^{n - 2} e^{- x^{2}} d x .

I_{n} = \frac{n - 1}{2} I_{n - 2} .

This is the full-line Gaussian moment recurrence; its initial values differ from the half-line version.

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