Characteristic function#
Calculating the characteristic function from the pdf (continuous distribution)#
- ctx.cf_from pdf(x, cf)
where
ctxisdec,mpm, orgmp.Let \(X\) denote a continuous univariate random variable with probability density function (PDF) \(\text{pdf}_X(x)\), cumulative distribution function (CDF) \(\text{cdf}_X(x)\), and moment generating function \(M_X(t)\). The characteristic function of the distribution of \(X\), given by the Fourier transform of its PDF, is defined as
\[C_X(t) = \operatorname{E} \left [e^{itX} \right] = M_X(it) = \int_{-\infty}^{\infty} e^{itx} \text{pdf}_X(x) \mathrm{d} x.\]Note however that the characteristic function of a distribution always exists, even when the probability density function or moment-generating function do not.
Calculating the characteristic function from the pmf (lattice distribution)#
- ctx.cf_from pmf(x, cf)
where
ctxisdec,mpm, orgmp.For a discrete distribution on the nonnegative integers, it is defined as
\[C_X(t) = \operatorname{E}[e^{itX}] = \sum_{j=0}^{\infty} e^{ijt} \text{pmf}(j)\]See Johnson(2005), p. 50
Calculating the characteristic function from the quantile function#
- ctx.cf_from qtf(x, cf)
where
ctxisdec,mpm, orgmp.\[C_X(t) = \operatorname{E} \left[ e^{itX} \right] = \int_{0}^{1} e^{it \: \text{qtf}_X(p)} dp.\]
Calculating the characteristic function from the raw moments#
- ctx.cf_from_rawmoments(x, cf)#
where
ctxisdec,mpm, orgmp.Formally, we can write
\[C_X(t) = 1 + \sum_{k=1}^{\infty} \mu'_X(k) \frac{i^k t^k}{k!}\]\[C_X(t) = 1 + \sum_{k=1}^{\infty} \operatorname {E} \left[X^{k}\right] \frac{i^k t^k}{k!}\]See also http://mathworld.wolfram.com/RawMoment.html for an expression involving cumulants.