09Chap2 - Transformation and Expectation 1 Function of a...

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Unformatted text preview: Transformation and Expectation 1 Function of a random variable Assume that X is a random variable with pmf/pdf f X , cdf F X . Denote the sample space of X by X . Then any function of X , say Y = g ( X ), is also a random variable. Examples: Y = X , X + 5, 3 X 2 , e- X , | X | , I { X > } . Denote the sample space of Y by Y . So g : X Y . Question : How to determine the distribution of Y ? For any subset A Y , we have P ( Y A ) = P ( g ( X ) A ) = P ( { x X : g ( x ) A } ) = P ( X g- 1 ( A )) , where g- 1 ( A ) = { x X : g ( x ) A } . Note that g- 1 ( A ) is the set of points in X that g ( x ) takes into the set A g- 1 is an inverse mapping from subsets of Y to subsets of X . It can be defined for any function g ( g is not necessarily one-to-one and/or onto). 2 Transformation of Discrete X Assume X is discrete with the pmf f X ( x ) = P ( X = x ). Let Y = g ( X ), then the sample space of Y is Y = { y : y = g ( x ) ,x X} . Since X is countable, so is Y . Therefore, Y is also a discrete random variable. 27 The pmf of Y can be computed as following f Y ( y ) = P ( Y = y ) = P ( g ( X ) = y ) = P ( X g- 1 ( { y } )) = X x g- 1 ( y ) f X ( x ) , for any y Y . Example. The distribution of X is x- 2- 1 1 2 f X ( x ) 0.1 0.2 0.4 0.2 0.1 Y = | X | . Find the pmf of Y . Example : ( Binomial Transformation ) Toss a coin n times whose probability of head is p . X is the number of heads. Then X has a binomial distribution, denoted as Bin( n,p ), with the pmf f X ( x ) = P ( X = x ) = n x p x (1- p ) n- x , x = 0 , ,n. Let Y denote the number of tails, i.e., Y = n- X . Find the pmf of Y . 28 3 Transformation of Continuous X Assume that both X and Y are continuous. It is convenient to define X = { x : f X ( x ) > } , Y = { y : y = g ( x ) for some x X} . The set { x : f X ( x ) > } is the support set of X . The cdf of Y = g ( X ) is F Y ( y ) = P ( Y y ) = P ( g ( X ) y ) = P ( { x X : g ( x ) y } ) = Z { x X : g ( x ) y } f X ( x ) dx. Example : ( Uniform Transformation ) Suppose X has a uniform distri- bution on the interval (0 , 2 ). Let Y = sin 2 ( X ). Describe the cdf of Y . 29 3.1 g Strictly Monotone (Increasing or Decreasing) Theorem. Let X F X with support X . Let Y = g ( X ) F Y with the sample space Y . (a) If g is increasing, then F Y ( y ) = F X ( g- 1 ( y )). (b) If g is decreasing, then F Y ( y ) = 1- F X ( g- 1 ( y ))....
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09Chap2 - Transformation and Expectation 1 Function of a...

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