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Finding this distribution

  • Thread starter Qyzren
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  • #1
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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!
 
Last edited:

Answers and Replies

  • #2
andrewkirk
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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.
 

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