Characteristic Function of Standard Gaussian Random Variable

Last updated Nov 1, 2022

Suppose $(\Omega, \mathcal{M}, \mathbb{P})$ is a Probability Space and $X$ is a Standard Gaussian Random Variable. Then

$$\phi_{X}(t) = \exp (\frac{t^{2}}{2})$$

Since $X$ is a Standard Gaussian Random Variable, we know $X \sim \mathcal{N}(0, 1)$. Applying the formula for the Characteristic Function of a Gaussian Random Variable:

$$\begin{align*} \phi_{X}(t) &= \exp (i t 0 - \frac{1^{2} t^{2}}{2})\\ &=\exp (\frac{t^{2}}{2}) \end{align*}$$ $\blacksquare$