Help Please! Stats query.

My question is based on the ActEd notes.

On page 90 there's a formula for using the exponential distribution to calculate reinsurance.

Retention level = £5,000
Percentage paid by reinsurer = 75% (of the excess over £5,000)
Total number of actual claims = 347
Lambda = 0.000407
Mean = 1/lambda

Expected total recoveries = 75% * 347 * (1/lambda) * exp(-lambda*5,000)

I'm having trouble understanding that last calculation. Yes, you guessed it, my stats is rusty!

 

My acted notes seem to have a differenet numbering.

There are a few projects which give this formula but I can't remember them offhand.

The instruction for the worked example has it. Its the integral from a to infinity of (x-a) times the pdf of the exponential (bits with lambda and e) equal to 1 over lambda times e to the minus lambda times a.

This gives the mean excess over the amount a.

The formula you quote is just the above times the proportion covered and the number of claims

 

"mean excess over amount a"

So does this include zero values? e.g. if there were 9 under £5,000 and 1 of £15,000; would the answer be £10,000 or £1,000?

I'm having real trouble getting sensible answers out of my spreadsheet.

 

Ahhh, just had a breakthrough (after a couple of pints, so might not cuont as a breakthrough in the morning). Thanks for your help.

 

OneLastTheorem, I guess you no longer need clarification but for anyone else who does.

It does include zero values. For your example the total excess over 5,000 would be 10,000, with the average being 1,000. This is because the excess over 5,000 are one value of 10,000 and nine values of 0.

For anyone wanting to look at the maths, it's the integral of

Max(0,x-a) *pdf of X over all values; which breaks down into

integral of 0 from -infinity (or zero or whatever) to a; plus
integral of (x-a) from a to infinity.

The first term vanishes.

ps I suppose that it would have been better to use "expected excess" than "mean excess"