Finding this distribution

Homework Statement

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.

Homework Equations

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!

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andrewkirk
Homework Helper
Gold Member
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.