I am working on the following task:
Let $X_1, X_2$ be two continuous, stochastic independence random variables and $X_i$~$Exp(\lambda_i)$ with $\lambda_1>\lambda_2>0$. Futhermore define $Z:=X_1+X_2$.
Determine the density function $g:=f^{X_1,Z}$ of $P^{X_1,X_2}:(\mathbb{R}^2,B)\to\mathbb{R}$ defined by $P^{(X_1,X_2)}(A)=\int_A gd\lambda^2$ and $A:=${$(s,t)\in\mathbb{R}^2|s\leq x_1, t+s\leq z$}.
So far I got the following results: The density function of $Z$ is given by $$f^Z(z)=\frac{\lambda_1\lambda_2}{\lambda_2-\lambda_1}(e^{-\lambda_1 z}-e^{-\lambda_2 z})\cdot 1_{(0,\infty)}(z).$$ Using this I determined the denisty function $g$ $$g(x_1,z)=f^{(X_1,Z)}(x_1,z)=f^{X_1}(x_1)\cdot f^{Z}(z)=\frac{\lambda_1^2\lambda_2}{\lambda_2-\lambda_1}e^{-\lambda_1 x_1}(e^{-\lambda_1 z}-e^{-\lambda_2 z})\cdot 1_{(0,\infty)^2}(x_1,z).$$
But I don't get how I need the definition of $P$ and especially $A$ for this task. Thanks for your answers!
(I'd write a comment but I can't since I don't have enough reputation)
Are you sure this is exactly the task? the definition of $A$ suggests that it is a random set that depends on $X_1$ and $Z$. Otherwise, since $z$ and $x_1$ are nowhere defined, $A$ wouldn't be well defined. But in this case, since $A$ would be random set, we have $A \not\in B$ and $P^{X_1,X_2}$ is not well defined. Maybe it should be something along the lines of:
$$P^{(X_1,X_2)}(C) = \int_C \int_{A(x_1, x_1 +x_2)}g d\lambda^2 d\mu(x_1,x_2) $$
where $\mu$ is the product measure of $X_1$ and $X_2$, $C\in B$ and I imposed two arguments into $A$ since I assume here that it is a random set depending on the two variables.