Let X1, X2, ..., Xn be iid random variables with continuous CDF FX and suppose the common mean is E(Xi) = μ. Define random variables Y1, Y2, ..., Yn by
Yi = 1 if Xi > μ; 0 if Xi ≤ μ. Find the distribution of ∑ni=1Yi.
I'm having a hard time figuring out how to begin to find the distribution.
Possibly Yn = (∑Xi - nμ ) /√n σ ?
The Attempt at a Solution
I'm having a hard time knowing where to begin...
If the question was p instead of μ, then Xi ~ Bernoulli(p) and the sum of n iid Bernoulli(p) is Binomial(n, p).
But since we have μ and not p, I'm also thinking that the central limit theorem tells us the sum of any random variable is always yields a normal distribution, with mean nμ? and variance ...?
Any help to get me in the right direction would be greatly appreciated!