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

Table of contents
- Solution ii

Solution

For the standard normal density,

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

. Using integration by parts,

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

, yielding the Gaussian fourth moment

E [X^{4}] = 3.

 Ancestors (13)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version