insurance exponential i would like to ask for help with following example, i know how to derive next steps that I am not even showing here, but cant derive the loglike function, thank you.

 A: You're dealing with the distribution of Surco's net payment amount*.  This is effectively a censored random variable. You need to write down the distribution of what Surco's costs are for the claims below 1000, and for those above 1000), and hence write down the log-likelihood.
*(for some unfathomable reason they don't seem to know the full claim size - the ground-up liability - only what they pay net of reinsurance ... but then how could they make claims on their excess of loss reinsurance without telling the reinsurer what the full claim amount was? Nevertheless, that's what you're dealing with in this question.)
Note that for claims below 1000 you have a density but for those above 1000 their net amount is always 1000 (you can calculate the probability that they pay 1000 rather than something less than it), so Surco's loss is a mixed (continuous and discrete) distribution. Label your observations such that the first 68 claims are below 1000 and then remainder are above it. Write the likelihood for the continuous and discrete parts separately and then use independence to put them all in one likelihood.
The uncensored and (right-)censored components of likelihood you have are the same as those discussed in the Wikipedia article on survival analysis, in the section on parameter estimation ("fitting parameters to data").
Note that the claims that exceed 1000 all yield payments of 1000. The probability that a claim exceeds 1000 is just $e^{-1000\lambda}$, the probability of observing a payment of 1000 -- which is then the contribution to the likelihood of those censored observations.
