= Characteristic function
{title2=$\varphi_X(t)$}
{wiki=Characteristic_function_(probability_theory)}
The characteristic function of a real <random variable> $X$ is
$$
\varphi_X(t)=\mathbb E[e^{itX}].
$$
It determines the <probability distribution> of $X$.
Here $t\in\mathbb R$ and $i^2=-1$. By <Euler's formula>, its <real part> and <imaginary part> are $\mathbb E[\cos(tX)]$ and $\mathbb E[\sin(tX)]$. The <complex exponential> has absolute value one, so this <expected value> exists for every real <random variable>, even without finite <absolute moments>. Moreover,
$$
\varphi_X(0)=1,\qquad |\varphi_X(t)|\leq\mathbb E[|e^{itX}|]=1,\qquad
\varphi_X(-t)=\overline{\varphi_X(t)}.
$$
The bound uses the <triangle inequality for a vector-valued integral>; the last identity is <complex conjugate> symmetry.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[§3.3.1 definition, printed p. 125; Theorem 3.3.1(a)–(c), printed p. 126]{ref} Consequently the <real part> is an <even function> and the <imaginary part> is an <odd function>.
For a <discrete probability distribution> with values $x_j$ and <probabilities> $p_j$, the <characteristic function> is $\sum_jp_je^{itx_j}$: a weighted average of points on the <unit circle> in the <complex plane>. For example, if $B$ has the <Bernoulli distribution> with $\mathbb P(B=1)=1/3$ and $\mathbb P(B=0)=2/3$, then
$$
\varphi_B(t)=\frac23+\frac13e^{it},\qquad
\varphi_B(2\pi/3)=\frac12+\frac{\sqrt3}{6}i,\qquad
|\varphi_B(2\pi/3)|=\frac1{\sqrt3}.
$$
Each point being averaged has absolute value one, but their directions partially cancel. At $t=\pi$, the same average is $1/3$; at $t=2\pi$, it returns to one. The <characteristic function> is thus neither a <probability density function> nor a <probability mass function>: its argument is a frequency parameter, and its values can be <complex numbers>. This <Bernoulli distribution> has no <probability density function> with respect to <Lebesgue measure>, yet its <characteristic function> is defined at every frequency.
\Image[/characteristic-function-weighted-phasors.png]
{title=A Bernoulli characteristic function as a weighted average on the unit circle}
{description=At $t=2\pi/3$, the two <complex exponentials> are $1$ and $e^{it}$. Left: their weighted average lies one third of the way along their connecting segment. Right: adding the weighted arrows $2/3$ and $e^{it}/3$ gives $\varphi_B(t)=1/2+i\sqrt3/6$. The dashed <unit circle> makes the absolute bound one visible.}
For a <probability density function> $f$, the same definition becomes $\varphi_X(t)=\int_{\mathbb R}e^{itx}f(x)\,dx$, the <Fourier transform> with the displayed sign convention. The <Laplace transform of a nonnegative random variable> instead uses $e^{-sX}$ for $s\geq0$, a real decaying weight. For the <Bernoulli distribution> above, it is $2/3+e^{-s}/3$; neither transform is the <probability density function>. The <uniqueness theorem for characteristic functions> explains why the oscillating averages still retain the entire <probability law>, while the <characteristic function of a sum of independent variables> turns combining laws into multiplication. These two properties make the representation useful even when directly computing a sum's <probability distribution> is cumbersome.
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