Central Limit Theorem

• Asymptotic analysis

• Central Limit Theorem (CLT):
  • sample size가 크다면 distribution of averages of iid variables (properly normalized) = a standard normal distribution

  • Normalization: ( X - μ ) / ( σ / sqrt(n) ) = Z ~ N( 0, 1 )

    • When population SD σ is unknown, the standard error of the sample mean (SEM) = SD of those sample means over all possible samples

    • SEM = SD / sqrt(n)

    • σ / sqrt ( n ) = the standard error of the sample mean

• Suppose X_1, X_2, ... X_n are independent, identically distributed random variables from an infinite population with mean μ and variance σ².

  • if n is large, the mean of the X's, call it X', is approximately normal with mean μ and variance σ² / n.

  • 則, X'~ N( μ, σ² / n ).

• 예시1: 동전 던지기, 앞면 확률 P(h)= p 라면, E(h) = p, variance = p ( 1 - p )

  • 동전 던지기를 n 번씩 했을 때, 그 결과의 평균값이 μ이라면,

  • normalization: ( μ - p ) / SD = ( μ - p ) / sqrt ( p ( 1 - p ) / n )