# Probability of Ruin at the first claim

The number of claims $$n \sim Po(\lambda)$$, and let $$X_n$$ denotes the claim amounts of a claim which are all iid and they follow a $$Exp(1)$$ distribution.

Assume the initial surplus is $$U$$, and the premium loading factor is $$\theta$$.

I am trying to find the probability of ruin for the first claim.

Here is what I have got:

Let $$T$$ denotes the time until the first claim and $$T\sim Exp(\lambda)$$, now I am looking for $$\psi(U,T)$$

and $$\psi(U,T) = \Pr(U + (1+\theta)\lambda T < X_1)= \Pr\Big(T < \frac{X_1 - U}{(1+\theta)\lambda}\Big)$$.

But now I am stuck, I am not sure how to continue with this expression.