NotesOct18 - p X then Y = T ( X ) is also discrete. The pmf...

Info iconThis preview shows pages 1–9. Sign up to view the full content.

View Full Document Right Arrow Icon
Transformations of random variables Statement of the problem: let X be a ran- dom variable with a known distribution, e.g. known cdf, or known pmf, or known pdf. Let Y = T ( X ) be a function, or a transforma- tion, of X . Find the distribution of Y .
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Computing the cdf of Y = T ( X ) If X is discrete with pmf p X , then F Y ( y ) = P ( Y y ) = P ( T ( X ) y ) = X x i : T ( x i ) y p X ( x i ) , -∞ < y < . If X is continuous with pdf f X then F Y ( y ) = P ( Y y ) = P ( T ( X ) y ) = Z x : T ( x ) y f X ( x ) dx, -∞ < y < .
Background image of page 2
Example : Suppose that X is uniformly dis- tributed between - 1 and 1. Find the distribu- tion of Y = X 2 .
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This method sometimes works even for trans- formations of several random variables. Example : Let X and Y be two indepen- dent standard exponential random variables, and Z = T ( X,Y ) = X + Y . Find the distribution of Z .
Background image of page 4
Computing the pmf or pdf of Y = T ( X ) . If the computation of the cdf of Y = T ( X ) is not practical, then in the discrete case, one can try computing the pmf of Y = T ( X ); in the continuous case one can try com- puting the pdf of Y = T ( X ).
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
If X is discrete with a probability mass function
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Background image of page 8
Background image of page 9
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: p X then Y = T ( X ) is also discrete. The pmf of Y : p Y ( y j ) = P ( Y = y j ) = P ( T ( X ) = y j ) = X x i : T ( x i )= y j p X ( x i ) . If the function T is one-to-one , then p Y ( y j ) = p X T-1 ( y j ) ( T-1 is the inverse map ) Example Let X be a mean Poisson random variable. Find the distribution of Y = 2 X . Let X be continuous with pdf f X . Sometimes Y = T ( X ) is also continuous. To compute the density of Y adjust by the derivative of the inverse transfor-mation . Monotone transformations : the function T is either increasing or decreasing on the range of X . A monotone function T is automatically one-to-one. The pdf of Y : f Y ( y ) = f X T-1 ( y ) dT-1 ( y ) dy ....
View Full Document

This note was uploaded on 12/03/2010 for the course OR&IE 3500 taught by Professor Samorodnitsky during the Fall '10 term at Cornell University (Engineering School).

Page1 / 9

NotesOct18 - p X then Y = T ( X ) is also discrete. The pmf...

This preview shows document pages 1 - 9. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online