- #1

Qyzren

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## Homework Statement

Let X

_{1}, X

_{2}, ..., X

_{n}be iid random variables with continuous CDF F

_{X}and suppose the common mean is E(X

_{i}) = μ. Define random variables Y

_{1}, Y

_{2}, ..., Y

_{n}by

Y

_{i}= 1 if X

_{i}> μ; 0 if X

_{i}≤ μ. Find the distribution of ∑

^{n}

_{i=1}Y

_{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!

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