Let $X$,$Y$,$Z$ be independent random variables with exponential distribution of parameter $\lambda$, then $X,Y,Z$ ~ $\xi(\lambda)$. The task is to calculate $P(X+Z>Y)$.
Comment: In previous excersices, by finding the joint density function of $X,Y$ ($f_{XY}=\lambda^{2}e^{-\lambda x}e^{-\lambda y}1_{[0,+\infty)x[0,+\infty)}(x,y)$) and integrating I got $P(X>Y)=\frac{1}{2}$ but I can't add $Z$ into the mix.
Edit: just in case, I removed what should be the answer.
By the same principle, you must triple integrate over the domain $\{(x,z,y):x\in[0{..}\infty), z\in[0{..}\infty),y\in[0{..}x+z)\}$
$$\mathsf P(X+Z>Y)=\int_0^\infty\int_0^\infty\int_0^{x+z} \lambda^3\mathrm e^{-\lambda(x+y+z)}~\mathrm d y~\mathrm d z~\mathrm d x $$