 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/i/solution

The standard normal density is even, so

x^{3} P (x)

is odd. Since Gaussian decay dominates every power, the integrand is absolutely integrable. Therefore

E [X^{3}] = \int_{- \infty}^{\infty} x^{3} P (x) d x = 0.

 New to topics? Read the docs here!