I am looking at Central Limit Theorem. It says that for a series of random sample W_1,...,W_2 under the same mean mu and standard deviation sigm, then the random variable (hat W - mu)/(sigma/sqrt(n)) follows normal distribution. However, it seems like for Binomial samples, (hat W - mu)/(sigma)=(hat W - mu)/(sqrt(np(1−p))) is normal. I want to know where the n−−√ term in the denominator went?

akuzativo617

akuzativo617

Answered question

2022-11-10

I am looking at Central Limit Theorem. It says that for a series of random sample W 1 , , W 2 under the same mean μ and standard deviation σ, then the random variable
W ¯ μ σ / n
follows normal distribution. However, it seems like for Binomial samples,
W ¯ μ σ = W ¯ n p n p ( 1 p )
is normal. I want to know where the n−−√ term in the denominator went?

Answer & Explanation

AtticaPlotowvi

AtticaPlotowvi

Beginner2022-11-11Added 18 answers

Since the binomial random variable W is the sum of n independent Bernoulli variables, the expression from CLT which is close to the standard normal distribution can be formulated for just one W:
W n p n p ( 1 p ) = W / n p p ( 1 p ) / n .
Here you can interpret W = X 1 + + X n where X i - i.i.d. Bernoulli variables. For the series of X i the expression becomes pretty standard:
X 1 + + X n n p n p ( 1 p ) = X ¯ p p ( 1 p ) / n .

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