Expectations and moment generating.pdf - PSTAT 120B Winter 2016 Lecture 1 or 2 supplement Expectations and moment generating functions Expected values

Expectations and moment generating.pdf - PSTAT 120B Winter...

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PSTAT 120B, Winter 2016, Lecture 1 or 2 supplement. Expectations and moment generating functions Expected values: The expected value of a random variable X is denoted E ( X ), or sometimes μ X or μ . If X is discrete, then E ( X ) = X all x x P ( X = x ) = X all x x f X ( x ), provided X all x | x | f X ( x ) < . If X is continuous with pdf f X ( x ), then E ( X ) = Z -∞ xf X ( x ) dx , provided Z -∞ | x | f X ( x ) dx < . Expectations of functions of random variables: Consider one random variable X . Let g ( X ) be any function of X . If X is a discrete random variable, then E [ g ( X )] = X all x g ( x ) P ( X = x ), provided X all x | g ( x ) | P ( X = x ) < . If X is a continuous random variable, then E [ g ( X )] = Z -∞ g ( x ) f X ( x ) dx , provided Z -∞ | g ( x ) | f X ( x ) dx < . Definition: Let X be any random variable with pdf f X ( x ). For any positive integer r , 1. The r th moment of X about the origin, is given by E ( X r ), provided this expected value exists. Special case: r = 1, gives E ( X ) = μ . 2. The r th moment of X about the mean, ( r th central moment), is defined as E [ ( X - μ ) r ] , where μ = E ( X ) , provided these moments exist .

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