Let
$R(t) = u + ct - \sum_{k=1}^{X(t)}Z_{k}, t\geq 0 $,
be a risk process, where $u> 0$ is the initial capital of the insurance company and $c> 0$ is a premium rate. We know that the number of claims received by the insurance company is in accordance with a homogenous Poisson Process $X(t), t\geq 0$, with parameter (intensity) $\lambda > 0$. Assume that the amounts $Z_k, k = 1, 2, ...$ of successive claims are independent random variables with common probability density function $f_{Z_k}(z), z \geq 0$. Suppose that random variables $Z_{k}, k = 1, 2, ...$ are independent of $X(t)$.
I would first like to be able to compute the ranges of $q, q \geq 0$ such that the function $f(z)=q\alpha \exp (-\alpha z) + (1 - q)\beta \exp (-\beta z); \beta > \alpha > 0,$ may be taken as $f_{Z_k}(z), z \geq 0$ in the risk process.
Then, I would like to be able to compute $E[R(t)]$, and $Var[R(t)]$ in terms of the parameters $u, c, \lambda, \alpha, \beta$ and $q$.
Can anyone help? I'm pretty stuck with this problem.
I'll start with some suggestions that should work. If you're still stuck after trying them, just comment under my answer, and I'll give more detail.
For the range of $q$, it sounds like you want to find the values of $q$ that make $f(z)$ a legitimate probability density function. If so, you want to find the range of $q$ values for which $f(z) \geq 0$ and $\int_0^{\infty} f(z) dz = 1$. That should be doable.
For $E[R(t)]$, use the fact that $E[R(t)] = E[E[R(t)|X(t)]]$. Knowing the distribution of $X(t)$ should make this doable as well.
For $Var[R(t)]$, try $Var[R(t)] = E[Var[R(t)|X(t)]] + Var[E[R(t)|X(t)]]$.