This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 onetoone 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 π ). 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 ))....
View
Full Document
 Spring '10
 weining
 Statistics, Variance, Probability theory

Click to edit the document details