Given that one knows the characteristic function of a rv $X$ how can we write the characteristic function of a function of $X$ when that function is $$\frac{1}{\sqrt{X+b}},$$ for $b$ constant? So, if I know the CF of $X$ to be some function $\phi_X=\mathbb E[e^{iux}]$ how do we find the characteristic function $$\phi_{\hat{X}}\mathbb =E\left[e^{iu\frac{1}{\sqrt{X+b}}}\right]$$ as a function of $\phi_X$? Is it just the manipulation of the exponential? If so, any suggestions?
2026-03-29 23:43:25.1774827805
Characteristic function of $a/\sqrt(X+b)$ given characteristic function of $X$
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