I have $P=X$ and the linear transformation $Q=X+c$ where $X\sim \operatorname{Ber}(p)$, $p \in (0,1), c\in R$.
To get the Total Variation (TV) I use the general formula
$$TV(P,Q) = \frac12\sum_{x\in E} |p_{\theta}(x) - p_{\theta'}(x)|$$
This yields the two pmfs
$$p'(x) = p^x(1-p)^{1-x}$$
$$q'(x) = p^{x-c}(1-p)^{1-x+c}$$
I am looking at the case where $c=1$ or $c=-1$ (since absolute, I guess no difference)
How do I best proceed?
When I input the two cases when $x=0$ and $x=1$ I always get the terms cancel out to $0$. ($p-p$ or $(1-p)-(1-p)$)