 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/c/ii/solution

For the standard normal density,

P^{'} (x) = - x P (x)

E [X^{4}] = - \int_{- \infty}^{\infty} x^{3} P^{'} (x) d x = - [x^{3} P (x)]_{- \infty}^{\infty} + 3 \int_{- \infty}^{\infty} x^{2} P (x) d x .

The boundary term vanishes because Gaussian decay dominates every power. Since the given mean is zero and variance is one,

E [X^{2}] = 1

E [X^{4}] = 3.

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