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app_bPROBABILITYSECOND - ECON 6002 Econometrics Memorial...

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ECON 6002 Econometrics Memorial University of Newfoundland Adapted from Vera Tabakova’s notes SECOND
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B.4.1 Mean, median and mode For a discrete random variable the expected value is: ( 29 ( 29 ( 29 1 1 2 2 [ ] n n E X x P X x x P X x x P X x = = + = + + = L 1 1 2 2 1 [ ] ( ) ( ) ( ) ( ) ( ) n n n i i i x E X x f x x f x x f x x f x xf x = μ = = + + + = = L Where f is the discrete PDF of x
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For a continuous random variable the expected value is: The mean has a flaw as a measure of the center of a probability distribution in that it can be pulled by extreme values. [ ] ( 29 E X xf x dx -∞ μ = =
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For a continuous distribution the median of X is the value m such that In symmetric distributions, like the familiar “bell-shaped curve” of the normal distribution, the mean and median are equal. The mode is the value of X at which the pdf is highest. ( 29 ( ) .5 P X m P X m = < =
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[ ( )] ( ) ( ) x E g X g x f x = [ ] [ ] E aX aE X = ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 E g X g x f x axf x a xf x aE X  = = = = Where g is any function of x, in particular;
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[ ] [ ] E aX b aE X b + = + ( 29 ( 29 ( 29 ( 29 1 2 1 2 E g X g X E g X E g X +  =  +
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The variance of a discrete or continuous random variable X is the expected value of ( 29 ( 29 2 g X X E X =  - The variance
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The variance of a random variable is important in characterizing the scale of measurement, and the spread of the probability distribution.
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