The capital $C$ of a bank grows proportionally with time: $$dC = adt$$ Therefore at time $t$, the bank has $at$ units of capital. The bank has to undergo stress tests, which occur according to a homogeneous Poisson process with intensity $\lambda$. The bank passes a stress test provided that it has at least a suffcient amount of capital at the moment of the test. The desired minimum amounts of required capitals are random, independent and following a continuous distribution with density
$$f(s)=\frac{4}{\pi(1+s^2)^2}.$$
What is the probability that the bank will pass the stress test ?
MY TRY
We can define $N(t)$ to be the number of failed tests up to time $t$. Then, one knows that $N(t)\sim\operatorname{Poisson}(p(t)\lambda)$ with $p(t)$ the probability to fail:
$$p(t)=\int_0^{\infty}\int_0^{s/a} \frac{4}{\pi(1+s^2)^2} dtds = \frac{2}{\pi a}$$
Then, I know $N(t)\sim\operatorname{Poisson}(\frac{2\lambda}{\pi a})$. But know I do not know why the answer should be $$\exp(-\frac{2\lambda}{\pi a}).$$
I was trying to compute $P(N(t)=0)=\exp(\frac{2\lambda}{\pi a} t)$ and then integrate this for all time $t$. But I do no get the same. Any help?