The Central Limit Theorem (CLT)

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Laplace (1810): Any sum or mean will - if the number of trials is large - be approximately normally distributed.

More precisely,
Suppose we have a random sample X₁, X₂,..., X_n from some underlying population with mean m and variance s². As long as s <¥ , for sufficiently large n,

the sum T₀=å_iX_i is approximately Normal with mean =nm and variance =ns²,
the mean =(å_iX_i)/n is approximately Normal with mean =m and variance =s²/n.

This result holds regardless of whether the original population is discrete or continuous, symmetric or skewed, unimodal or multimodal, or any has other characteristic. The quality of the approximation depends on the underlying distribution as well as on the sample size n. In most cases the approximation is adequate as long as n>30, and if n>100 the approximation is excellent for virtually any distribution.

Linear Combinations: Suppose X_i has mean m_i and S.D. s_i, i=1,2,...,n
    E[a₁X₁+a₂X₂+...+a_nX_n) = a₁m₁+a₂m₂+...+a_nm_n
    Var[a₁X₁+a₂X₂+...+a_nX_n) = å_iå_ja_ia_jCov(X_i,X_j)
                = a₁²s₁²+a₂²s₂²+...+a_n²s_n²if all X_i are independent of each other.
(NOTE: Cov(X_i,X_i)=Var(X_i)=s_i²)

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Jayant Rajgopal