Characteristic Function of Gaussian Distribution/Corollary
Jump to navigation
Jump to search
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.