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X is a real-valued random variable with characteristic function:

$\hspace{20mm}$$\phi(t) = \frac{1}{2}[\cos(t) + \cos(t\pi)]$, $\hspace{10mm}-\infty<t<\infty$.

  1. Is the distribution of X completely continuous?

  2. Is there an $r$ such that $E(X^r)$ does not exist or is infinite?

    1. The distribution is not absolutely continuous as the limit of $\sin(x)$ as $x$ approaches infinity does not exist.

    2. ?

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1 Answer 1

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By definition of the Fourier transform $\mathcal{F}$ (up to some convention-dependent constants),

$$ \delta_x \stackrel{\mathcal{F}}{\mapsto} e^{itx}. $$

So

$$ \frac{1}{2} (\delta_1 + \delta_{-1}) \stackrel{\mathcal{F}}{\mapsto} \cos t, $$

and

$$ \frac{1}{2} (\delta_{\pi} + \delta_{-\pi}) \stackrel{\mathcal{F}}{\mapsto} \cos \pi t, $$

So your $X$ induces the uniform measure supported on $\{1, -1, \pi, -\pi\}$.

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