I have a question that I encountered while reading my notes. Suppose we have two independent variables
\begin{align*} &P_X(dx) = e^{-x} \mathbb{1}_{\mathbb{R}_+}dx \\ &P_Y = \frac{1}{2}\sum\limits^{\infty}_{k=0} 2^{-k} \delta_k \end{align*}
I need to calculate $\mathbb{P}({X+Y \leq 3})$.
What I have tried: According to the examples with continuous variables, I thought that there's two way to calculate it. First of all we need to find $P_{(X,Y)}$ which is easy then we can either take measure on set $G=\{(x,y)\in \mathbb{R}^2 : x+y\leq 3\}$. Or calculate $P_{f(X,Y)}$, where $f(X,Y)=X+Y$. Both time I either fail to calculate the integrals or their bounds (I think since I don't know the answer, the problematic part is mixing discrete and continuous part). It would be nice if someone could show how to do it either way, so I could know what I do wrong.
homeworktag. You might want to delete the measure theory tag too since the use of formal measure theory here is really using a bomb to kill a fly. The probability that you want is most easile computed via the law of total probability as $$P\{X+Y \leq 3\} = \sum_{i=0}^3 P\{X \leq 3-i \mid Y = i\}P\{Y = i\}$$ – Dilip Sarwate May 16 '12 at 15:52