# How to calculate the mean of a sub sample, given the mean of the super sample and the standard deviation of the population?

I need to run a simulation of cash flows for a project. We are selling a service. The service comes with a range of options. Depending on the specific options chosen, the cost of the service can change. I have determined that the mean of the price of the service is 605 and the standard deviation is 317.

So if we expect to make 100 sales next month, the expected sample mean is obviously the same as the population mean, and the standard deviation of the mean is 317 / sqrt(100).

The service is charged monthly, and cancellations can happen from month to month. I have a good idea of the distribution of cancellation rates. For now let's assume they are constant at 5% per month. So in month 1, I sell 100, then of those 100, 5 get cancelled in month 2, of those 95, another 5% get cancelled in month 3 etc.

I want to work out the distribution of the recurring revenue for contracts in month n, given the mean revenue of contracts at n-1.

Another way of stating this: I have an infinite bucket of numbers with a known mean and standard deviation. I draw a sample of size 100 from that bucket. I only know the mean of this sample. I then draw a second sample of size 95 from the first sample. What is the distribution of the sample mean of the second sample, given the mean of the first sample and the population standard deviation?

UPDATE

Using a chi squared distribution, I can sample the standard deviation of the sample. So for sample 1, I know the mean and the standard deviation. But how to obtain the mean and standard deviation of sample 2?