I have this exercise:
Let :
$\begin{array}{lcl} X & \sim & Po(\lambda_1), \\ Y & \sim & Po(\lambda_2), \\ Z & \sim & Bin(1, \frac{1}{2}) \end{array}$
be independent random variables, where $\lambda_1, \lambda_2 > 0$.
Consider $T := (X+Y)Z-1$
Determine the codomine of $T$.
If $\lambda_1 = 1, \lambda_2 = 2$ determine the probability function of $T$.
I have tried but without success to do it. Here we have that:
$\begin{array}{lcl} X & \sim & Po(1), \\ Y & \sim & Po(2), \\ Z & \sim & Bin(1, \frac{1}{2}) \end{array}$
therefore, I have considered the probability function for each random var
$\begin{array}{lcl}P(X=x) & = & e^{-1} \frac{1^x}{x!} & \mbox{for} & x = 0, \dots, +\infty \\ P(Y = y) & = & e^{-2} \frac{2^y}{y!} & \mbox{for} & y = 0, \dots, +\infty \\ P(Z = z) & = & \binom{1}{z}(\frac{1}{2})^{1-z}(\frac{1}{2})^z & \mbox{for} & z = 0, \dots, 1 \end{array}$
but, from here, please can you tell me how can I continue? Thanks!