Characteristic Function

Topics: Probability

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(definition)

Let $X$ be a random variable.

The characteristic function of $X$, denoted ${\phi }_X:\mathbb{R}\to \mathbb{C}$, is defined as:

$${\phi }_X(t)=\mathbb{E}[e^{itX}]$$

…where $i=\sqrt{-1}$, of course.

Characteristic functions have several properties.

Discrete Case §

(theorem)

Knowing the form of the expected value for a random variable. the characteristic function of such a random variable is:

$${\phi }_X(t)=\sum _{x\in R_X}{e}^{itx}{f}_X(x)$$

Continuous Case §

(theorem)

Similarly, for an absolutely continuous random variable:

$$\begin{array}{rl}{\phi }_X(t)& ={\int }_{\mathbb{R}}{e}^{itx}{f}_X(x)dx\\ & ={\int }_{\mathbb{R}}{e}^{itx}d{f}_X(x)\end{array}$$

(theorem)

The characteristic function for $X$ is also given by:

$${\phi }_X(t)=\int \cos (tx)f_X(x)dx+i\int \sin (tx)f_X(x)dx$$

Where from?

Remember that $e^x$ can be expressed as a Taylor series as follows:

$$e^x=\sum _{n=0}^{\infty }\frac{x^n}{n!}=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\dots$$

Setting $x=iy$, with $y\in \mathbb{R}$, we get:

$$\begin{array}{rl}{e}^{iy}& =1+(iy)+\frac{(iy{)}^2}{2!}+\dots \\ & =\left(1-\frac{y^2}{2!}+\frac{y^4}{4!}-\dots \right)+i\left(y-\frac{y^3}{3!}+\frac{y^5}{5!}-\dots \right)\\ & =\cos (y)+i\sin (y)\end{array}$$

The latter equivalence holding true due to the representation of $\cos (y)$ and $\sin (y)$ as Taylor series.