Question
Show that $X$ and $-X$ have the same distribution if and only if $\phi_{X}$ is a purely real-valued function.
Step 1
First, let's recall the definition of the characteristic function of a random variable $X$: $$\phi_X(t) = E[e^{itX}] = \int_{-\infty}^{\infty} e^{itx} f_X(x) dx$$ where $f_X(x)$ is the probability density function (pdf) of $X$. Show more…
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