6.1.4 Characteristic Functions
There are random variables for which the moment generating function does not exist on any real interval with positive length. For example, consider the random variable $X$ that has a
Cauchy distribution
\begin{align}%\label{}
f_X(x)=\frac{\frac{1}{\pi}}{1+x^2}, \hspace{10pt} \textrm{for all }x \in \mathbb{R}.
\end{align}
You can show that for any nonzero real number $s$
\begin{align}%\label{}
\nonumber M_{X}(s)=\int_{-\infty}^{\infty} e^{sx}\frac{\frac{1}{\pi}}{1+x^2} dx=\infty.
\end{align}
Therefore, the moment generating function does not exist for this random variable on any real interval with positive length.
If a random variable does not have a well-defined MGF, we can use the characteristic function defined as
\begin{align}%\label{}
\phi_{X}(\omega)&=E[e^{j \omega X}],
\end{align}
where $j=\sqrt{-1}$ and $\omega$ is a real number. It is worth noting that $e^{j \omega X}$ is a complex-valued random variable. We have not discussed complex-valued random variables. Nevertheless, you can imagine that a complex random variable can be written as $X=Y+jZ$, where $Y$ and $Z$ are ordinary real-valued random variables. Thus, working with a complex random variable is like working with two real-valued random variables. The advantage of the characteristic function is that it is defined for all real-valued random variables. Specifically, if $X$ is a real-valued random variable, we can write
\begin{align}%\label{}
|e^{j \omega X}|=1.
\end{align}
Therefore, we conclude
\begin{align}%\label{}
|\phi_{X}(\omega)|&=|E[e^{j \omega X}]| \\
&\leq E[|e^{j \omega X}|]\\
&\leq 1.
\end{align}
The characteristic function has similar properties to the MGF. For example, if $X$ and $Y$ are independent
\begin{align}%\label{}
\phi_{X+Y}(\omega)&=E[e^{j \omega (X+Y)}]\\
&=E[e^{j \omega X} e^{j \omega Y}]\\
&=E[e^{j \omega X}]E[e^{j \omega Y}] \hspace{10pt} \textrm{(since $X$ and $Y$ are independent)}\\
&=\phi_{X}(\omega) \phi_{Y}(\omega).
\end{align}
More generally, if $X_1$, $X_2$, ..., $X_n$ are $n$
independent random variables, then
\begin{align}
\nonumber \phi_{X_1+X_2+\cdots +X_n}(\omega)=\phi_{X_1}(\omega) \phi_{X_2}(\omega) \cdots \phi_{X_n}(\omega).
\end{align}
Example
If $X \sim Exponential (\lambda)$, show that
\begin{align}%\label{}
\phi_{X}(\omega)&=\frac{\lambda}{\lambda-j\omega}.
\end{align}
- Solution
-
Recall that the PDF of $X$ is
\begin{align}%\label{}
\nonumber f_X(x)=\lambda e^{-\lambda x} u(x),
\end{align}
where $u(x)$ is the unit step function. We conclude
\begin{align}%\label{}
\nonumber \phi_{X}(\omega)&=E[e^{j \omega X}] \\
\nonumber &=\int_{0}^{\infty}\lambda e^{-\lambda x} e^{j \omega x}dx\\
&=\left[\frac{\lambda}{j \omega-\lambda} e^{(j \omega-\lambda) x}\right]_{0}^{\infty}\\
\nonumber &=\frac{\lambda}{\lambda-j \omega}.
\end{align}
Note that since $\lambda>0$, the value of $e^{(j \omega-\lambda) x}$, when evaluated at $x=+\infty$, is zero.
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Practical uncertainty: Useful Ideas in Decision-Making, Risk, Randomness, & AI
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