Let $X, Y \sim \mathrm{Exp}(\lambda)$. I want to show that $Z = X+Y$ and $W = X/(X+Y)$ are independent.
I know that $Z \sim \Gamma(2,\lambda)$ and that $W \sim \mathrm{Uniform}([0,1])$, but I do not know how I can show these distributions are independent of one another.
I know that they have moment generating functions $(\frac{\lambda}{\lambda-\theta})^2$ and $\frac{1}{\theta}e^\theta$ respectively, so unless I am missing something easier, I'm thinking that the way I can show they are independent is to show that $$\mathbb{E}[e^{\alpha Z + \beta W}] = \Bigm(\frac{\lambda}{\lambda-\alpha}\Bigm)^2 \cdot\,\, \frac{1}{\beta}e^\beta\text{.}$$
I'm not too familiar with MGFs and I'm not quite sure how I'd actually interpret the expression, $\mathbb{E}[e^{\alpha Z + \beta W}]$, on the left hand side, in terms of some integral I can manipulate into the right hand side.
How should I interpret an expression like this, and is there an easier way to show $Z$ and $W$ are independent?