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The sort of problem that I'm working on is actually related to the following previous post:

Bernoulli Distribution with support different from $\{0,1\}$

And if $X$ is Rademacher distributed then how do I compute the characteristic for $Y=X+c :(c\ge 0 $ $constant)$

I have the following: $φ_Y=φ_{X+c}=\cos(t)*\mathbb{E}[e^{itc}]$

I just get the feeling like this is "messy" and should some more elegant/useful representation.

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Since $c$ is constant, $\mathbb E(\mathrm e^{\mathrm itc})=\mathrm e^{\mathrm itc}$. (And please replace $\varphi_Y$ and $\varphi_{X+c}$ by $\varphi_Y(t)$ and $\varphi_{X+c}(t)$.)