Imagine tossing a fair coin 1,000 times for 10 consecutive days, giving 10,000 trials in total.
I understand that using binomial statistics the standard deviation of the number of heads would be:
sqrt(npq) = 50 (with a mean of 0.5 x 10,000 = 5,000)
But now, instead of the coin being fair, the underlying likelihood for the coin to land heads is now not fixed at 0.5, but it has a mean of 0.5 and a standard deviation of 0.1 - and this underlying likelihood changes every day.
Day 1 the underlying likelihood of getting a head is 0.58 Day 2 the underlying likelihood of getting a head is 0.49 Day 3 the underlying likelihood of getting a head is 0.51
(Maybe you are picking a new coin each day and they are "bad" with a fairness standard deviation of 0.1)
So there are now 2 standard deviations involved. One is related to the fairness of the coin and how it changes daily (which is 0.1). Secondly, there is the standard deviation expected from binomial statistics, which is
sqrt(npq)/n = 50/10,000 = 0.003.
My question, is how do I combine these two standard deviations. In other words, I guess the mean of my 10,000 trials should still be 5,000 heads. But what would the new standard deviation be?
Any help in how to think about this is much appreciated.