MHB Discrepency with book's answer to characteristic function

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The characteristic function for the PMF \(p_X[k] = \frac{1}{5}\) for \(k = -2, -1, \ldots, 2\) is derived using the formula \(\phi_X(\omega) = \sum_{k} e^{i\omega k} \mathbb{P}(X = k)\). This leads to the expression \(\frac{1}{5}(e^{-2i\omega} + e^{-i\omega} + 1 + e^{i\omega} + e^{2i\omega})\). By applying Euler's formula, the expression can be simplified to \(\frac{1}{5}(1 + 2\cos(\omega) + 2\cos(2\omega))\). The discrepancy with the book's answer arises from a misunderstanding of the characteristic function's definition. Ultimately, the correct characteristic function aligns with the book's result when properly simplified.
Dustinsfl
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Find the characteristic function for the PMF \(p_X[k] = \frac{1}{5}\) for \(k = -2, -1,\ldots, 2\).

The characteristic function can be found with
\begin{align*}
\phi_X(\omega) &= E[\exp(i\omega X)]\\
&= \frac{1}{5}\sum_kke^{i\omega k}\\
&= \frac{1}{5}\big(-2e^{-2i\omega} - e^{-i\omega} +
2e^{2i\omega} + e^{i\omega}\big)\\
&= \frac{2i}{5}\bigg(\frac{e^{i\omega} - e^{-i\omega}}{2i} +
\frac{e^{2i\omega} - e^{-2i\omega}}{i}\bigg)\\
&= \frac{2i}{5}\big(1 + 4\cos(x)\big)\sin(\omega)
\end{align*}

However, the book says the answer is \(\frac{1}{5}(1 + 2\cos(\omega) + 2\cos(2\omega))\).
 
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The problem is your definition of the characteristic function. For a discrete random variable the characteristic function is defined as
$$\phi_X(\omega) = \sum_{k} e^{i\omega k} \mathbb{P}(X = k) $$
in this case we get
$$= \frac{1}{5} \sum_{k} e^{i \omega k} = \frac{1}{5} \left(e^{-2i\omega}+e^{-i\omega}+1+e^{i \omega}+e^{2i\omega}\right)$$

Now, use the fact that $e^{i \omega} = \cos \omega + i \sin \omega$ and simplify.
 
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