Moments

# Moments - dna 0= E = Z E nehT = Z teL = X X(E X 2 SELBAIRAV...

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Moments and Moment Generating Functions 1 M OMENTS OF R ANDOM V ARIABLES M EAN The mean measures the weighted average of the realizations of the random variable, () x X E Xx f x d x μ == A . V ARIANCE The variance measures of the weighted average of the squared deviations from the mean, 2 22 ( ) x X E f x d x σμ μ ⎡⎤ =− = ⎣⎦ A The formula for the variance squares the deviations, so that the positive and the negative deviations of the same absolute size contribute to the variance in the same way, although if the density function is not symmetric, then they are weighted differently. It is worth noting that 2 2 2 1 2 ( ) (2 ) ( ) 2 2 2 . x x xx x x X X XX X EX X x fx d x f x d x x d x x d x d x x d x x d x d x μ μμ μ −= + + + + ∫∫ A A AA A A S TANDARDIZED R ANDOM V ARIABLES Let X Z μ σ = . Then 0 XE X EZ E σσ −− ⎛⎞ = ⎜⎟ ⎝⎠ and

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Moments and Moment Generating Functions 2 2 2 2 2 2 () 1 EX X EZ E μ μσ σ σσ 2 ⎡⎤ ⎛⎞ ⎣⎦ == = = ⎢⎥ ⎜⎟ ⎝⎠ . Note that this property holds for any random variable X with finite mean and variance. S KEWNESS In some distributions, positive deviations from the mean tend to be larger in magnitude than negative deviations, or vice versa. The skewness is a measure of the tendency of de- viations from the mean to be larger in one direction, 3 3 ( ) x X x fx d x μμ −= A Like we did with the variance, skewness can be expressed in terms of the first three raw moments of X , E ( X ), E ( X 2 ), and E ( X 3 ), 22 33 2 3 32 2 3 3 3 ( ) ( 3 3 ) ( ) 3 3 3 3 ( ) 3 . xx x x XX X X xf x d x x f x d x x f xdx f xdx d x μ σ μ μ μ μ μ σμ μ 2 + 2 + =− + + ∫∫ AA A A Because the deviations are cubed rather than squared, the signs of the deviations are maintained. Cubing the deviations emphasizes the effects of large deviations from the mean. Skewness can be positive or negative. Positively skewed distributions have rela- tively thick right-hand tails, while negatively skewed distributions have relatively thick left-hand tails. The coefficient of skewness is the third moment of the standardized random variable, ( x X Xx Ef x d x μ η σ 3 3 −) −− ⎛⎞⎛ ≡= = ⎜⎟⎜ ⎝⎠⎝ A
Moments and Moment Generating Functions 3 This unitless measure of the shape of the distribution determines whether the left (nega- tive or right tail of the distribution is thicker. The formula includes a divisor of 3 σ

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## This note was uploaded on 08/01/2008 for the course ARE 210 taught by Professor Lafrance during the Fall '07 term at Berkeley.

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Moments - dna 0= E = Z E nehT = Z teL = X X(E X 2 SELBAIRAV...

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