Find the pdf of y for all y 0 1 let x fx 1 y dfx x

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Unformatted text preview: a = 0 (otherwise it is easy), we have fY ( y ) = M. Chen (IE@CUHK) 1 fX |a| ENGG2430C lecture 9 y −b a 14 / 20 Example: FX (X ) Consider a continuous r.v. X with strict monotonic CDF FX (x ) Let Y = FX (X ). It is a random variable. Find the PDF of Y : for all − y ∈ [0, 1], let x = FX 1 (y ): dFX (x )| dx = fX (x )/fX (x ) fY (y ) = fX (x )/| =1 As such, fY ( y ) = 1, 0, if y ∈ [0, 1]; otherwise. For any r.v. X , Y = FX (X ) is a r.v. uniformly distributed in [0, 1] M. Chen (IE@CUHK) ENGG2430C lecture 9 15 / 20 − Example: FX 1 (U ) Where r.v. U ∼ U[0, 1] Let U be a random variable uniformly distributed in [0,1]. Let − Z = FX 1 (U ).Find the PDF and CDF of Z . − Z = FX 1 (U ), so U = FX (Z ). f U (u ) = fZ (z ) dFX | dz (z )| = fZ ( z ) fX (z ) U is a random variable uniformly distributed in [0,1], so 1 = f U (u ) = fZ (z ) . and f X (z ) fZ (z ) = fX (z ), and FZ (z ) = FX (z ). M. Chen (IE@CUHK) ENGG2430C lecture 9 16 / 20 Generating Random Variable Question: How to generate a discrete random variable X the following distribution? (From the textbook) M. Chen (IE@CUHK) ENGG2430C lecture 9 17 / 20 Generating Continuous Random Variable Given a strictly increasing CDF F (x ) and a random variable U uniformly distributed in [0,1], the procedure t...
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This document was uploaded on 03/31/2014.

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