Discussion
Suppose X follows Cauchy distribution with location parameter \(\mu\) and scale \(\sigma\). Then the characteristic function of X is :
Answer: A
The characteristic function is \(\exp\{it\mu - \sigma|t|\}\).
The standard Cauchy density \(1/(\pi(1 + x^{2}))\) has characteristic function \(e^{-|t|}\). For \(X = \mu + \sigma Z\), \(\phi_{X}(t) = e^{it\mu}\phi_{Z}(\sigma t) = e^{it\mu - \sigma|t|}\).
The absolute value matters: \(e^{it\mu - \sigma t}\) would have modulus \(e^{-\sigma t} \gt 1\) for \(t \lt 0\), impossible for a characteristic function. \(\exp\{it\mu - \sigma^{2}t^{2}/2\}\) is the normal characteristic function, and the last option has no meaning. The kink at \(t = 0\) reflects that the Cauchy distribution has no mean.
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