PMF function: joint pmf question For the PMF of $X$ 
$$P(X = x) = (0.5)^x \;\text{ for }\;x\in\mathbb{Z}^+,$$

would the PMF of $Y= X-1$ just be

$$P(Y = x) = (0.5)^{x-1}.$$
$$
\Pr(Y=x) = \Pr(X-1=x) = \Pr(X=x+1) = 0.5^{x+1}.
$$
if X is a continus random variable whith PMF f that f(x) > 0 for all x, supose that y=H(x) be a function strictly monotonous, so the PMF Y=H(x) is:
$$
g(y) = f(x)|\frac{\mathrm{d}x}{\mathrm{d}y}|
$$
Solving the question...
$$
\\ g(y)=x-1\\ x = y+1\\\frac{\mathrm{d}x}{\mathrm{d}y}=1\\g(y)=0.5^{y+1}
$$