What is the distribution of the sum of n iid Bernoulli random variables?

[Qyzren]

Homework Statement

Let X₁, X₂, ..., X_n be iid random variables with continuous CDF F_X and suppose the common mean is E(X_i) = μ. Define random variables Y₁, Y₂, ..., Y_n by
Y_i = 1 if X_i > μ; 0 if X_i ≤ μ. Find the distribution of ∑ⁿ_i=1Y_i.

I'm having a hard time figuring out how to begin to find the distribution.

Homework Equations

Possibly Y_n = (∑X_i - 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 X_i ~ 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!

 

[andrewkirk]

Define $p_i=Prob(X_i\le \mu)$. Since the $X_i$ are iid, $p_i$ is the same for all $i$ so we can just write it as $p$. Then each $Y_i$ is a Bernoulli.