Expected Value of the Cosine of a Random Variable Following the Standard Normal Distribution

September 28, 2020

Proposition

For a random variable $X$ following the standard normal distribution $N(0, 1)$, the expected value of $\cos{X}$ is $\frac{1}{\sqrt{e}}$.

Proof Sketch 1

Using the decomposition of $\cos{x} = \frac{e^{ix} + e^{-ix}}{2}$ and completing the square in the exponent. Although complex integration is involved, considering the integration path connecting the points $-R, -R \pm i, R \pm i, R$ forms a rectangular path. Using Cauchy’s integral theorem, taking the limit as $R \to \infty$, it can be shown that the integration is equivalent to integrating along the real axis.

Proof Sketch 2

Using the Maclaurin series expansion of $\cos{x}$ and performing term-by-term integration. By performing integration by parts, we obtain

$$ I_n := E[X^{2n}] = \int_{-\infty}^{\infty} x^{2n} \frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}}dx $$

then, we have

$$ I_{n+1} = (2n + 1) I_n = (2n + 1)!! I_0 = (2n + 1)!! $$

By utilizing this result, the Maclaurin series expansion of $e^{x}$ evaluated at $x = -\frac{1}{2}$ can be obtained.

Note

It should be noted that the expected value of $\sin{X}$ is trivial and equals $0$. This is because the integrand becomes the product of an even function and an odd function.