Chapter 7
Characteristic functions
Lectures 31 - 33
In this chapter, we introduce the notion of characteristic function of a random variable and study its properties. Characteristic function serves as an important tool for analyzing random phenomenon.
Definition 7.1 (Characteristic functions) The characteristic function of a random variable $ X$ is defined as
$\displaystyle \Phi_X(t) \ = \ Ee^{itX} , t \in \mathbb{R}. $
(where $ Ee^{itX} \ = \ E \cos tX + i E \sin tX)$
Example 7.0.43   Let $ X \thicksim {\rm Bernoulli}(p)$. Then
$\displaystyle \phi_X(t) \, = \, (1-p) + p e^{it} \, . $
Example 7.0.44   Let $ X \thicksim {\rm exponential} (\lambda)$. Then
\begin{displaymath} \begin{array}{lll} \Phi_X(t) \ = \ Ee^{itX} & = &\displays... ...bda+it)}{\lambda^2+t^2} , \, t \in \mathbb{R} }. \end{array} \end{displaymath}