Characteristic Function of Gaussian Distribution/Corollary

Theorem

The characteristic function of the standard Gaussian distribution is:

$\map \phi t = e^{-\frac 1 2 t^2}$

Proof

Recall Characteristic Function of Gaussian Distribution:

The characteristic function of the Gaussian distribution with mean $\mu$ and variance $\sigma^2$ is given by:

$\map \phi t = e^{i t \mu - \frac 1 2 t^2 \sigma^2}$

The standard Gaussian distribution is the Gaussian distribution with $\mu = 0$ and $\sigma = 1$.

Hence:

\(\ds \map \phi t\)

\(=\)

\(\ds e^{i t \times 0 - \frac 1 2 t^2 \times 1^2}\)

\(\ds \)

\(=\)

\(\ds e^{0 - \frac 1 2 t^2}\)

\(\ds \)

\(=\)

\(\ds e^{-\frac 1 2 t^2}\)

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): characteristic function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): characteristic function: 1.