STA305H1 Lecture Notes - Lecture 1: Kurtosis, Central Limit Theorem, Exponential Distribution
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18 Jan 2018
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Skewness and kurtosis and the central limit theorem. Skewness and kurtosis are, respectively, meaures of the symmetry and tail heaviness of a probability distribution. If x is a random variable with distribution f then the skewness and kurtosis of x (or of f ) are skew(x) = kurt(x) = 4 e[(x )4] where = e(x) and 2 = var(x). If x n ( , 2) then skew(x) = 0 (since the density of x is symmetric around its mean) and kurt(x) = 3. For example, if x has a uniform distribution on [a, b] then skew(x) = 0 and kurt(x) = 1/5 while if x has a exponential distribution then skew(x) = 2 and kurt(x) = 9. Thus we might expect that the skewness and kurtosis of sn will be approximately 0 and 3, respectively, when n is large. Suppose that x1, , xn are independent with e(x 4.