Let X(1),X(2),.....,X(n) be independent and identically distributed random variables each having mean u and standard deviation sigma^2 . Assume that we do not know the distribution of X(i) , n=1......n, Let Y = (X(1)+X(2)+....+X(n) ) / n  Prove that the expected value of the random variable Y is u. the variance of the random variable Y is sigma^2 / n What can be said about the distribution of Y when n is large

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Let X(1),X(2),.....,X(n) be independent and identically distributed random variables each having mean u and standard deviation sigma^2 . Assume that we do not know the distribution of X(i) , n=1......n,

Let Y = (X(1)+X(2)+....+X(n) ) / n 

Prove that

  1. the expected value of the random variable Y is u.
  2. the variance of the random variable Y is sigma^2 / n
  3. What can be said about the distribution of Y when n is large
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