Tools from Functions of One RV

Tools from Functions of One RV - NORMALIZATION For any...

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NORMALIZATION 2 0 For any random variable , discrete or continuous, with mean and variance , its normarlized random variable has zero mean and unit variance. X X X µσ µ σ 2 Review: ( ) () a Xba Xb VAR aX b a VAR X += + Homework is (0,1). Normalize . XU X

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GENERATING SAMPLES OF A RANDOM VARIABLE WITH A COMPUTER Computers have a built-in routine to generate (0,1), a uniformly distributed random number between 0 and 1. We want to generate a random number according to some pdf ( ). X U fx Then do the following: i) Plot the cdf ( ) ii) Generate from the uniform distribution iii) Find that produces ( ) X X Fx u r uFr = obtained from the above has pdf ( ) or the cdf ( ) X X R 1.0 0.0 u ) u ( F 1 X ) x ( F X 1 () X RF U = X UFR =
Why it works? [ ] { } -1 Let be the uniform random variable between 0 and 1 Define a new random variable ( ) Th en () P P r () Therefore and have the same cdf. X R XX U RFU F r Rr UFr Fr RX = = ≤= = Example: Generate samples of an exponential rv ( ) ( ) 1 0. Write ( ) 1 ln 1 Use as the exponential random variable.

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Tools from Functions of One RV - NORMALIZATION For any...

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