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
$$P(Y = x) = (0.5)^{x-1}.$$

The final random variable can be express as the sum of tow random variables $X$ and $Y$ defined as:

*

*$P(X = x) = 0.5^x$

*$Y(Y = y) = \delta(y+1)$, where $\delta$ is the Dirac Delta distribution

The final PMF is the inverse Laplace of product of Laplace transform of $X$ and $Y$:

*

*$PMF = \mathcal{L}^{-1}\{e^s\mathcal{L}\{0.5^x\}\}$.

So the final PMF is $P(x+1)u(t+1)$, where $u(t-a)$ is the Heaviside step function.

*

*$PMF = 0.5^{x+1}$ for all $x\gt-1$.

