What is characteristic function of normal distribution?

The characteristic function of the normal distribution is given by:

ϕ(t) = exp[itμ - ½σ²t²]

where μ is the mean and σ² is the variance of the normal distribution.

The characteristic function is a complex-valued function of the parameter t. It describes the distribution of a random variable in terms of its moment-generating function. The moment-generating function is the expected value of e^{tx}, where x is the random variable of interest. The characteristic function can be used to calculate moments of the distribution, such as the mean and variance.

The properties of the normal distribution's characteristic function include:

1. It is a continuous function of t.
2. It is infinitely differentiable.
3. It is symmetric about the origin.
4. The characteristic function for a sum of independent normal random variables is the product of their individual characteristic functions.
5. The inverse Fourier transform of the logarithm of the characteristic function gives the density function of the normal distribution.