# This way we say that we generate a random variable x

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Unformatted text preview: o generate an X is: Step 1: Generate a value of U using the computer, say the value is u . Step 2: Compute the value x of X which satisﬁes that F (x ) = u . Step 3: Repeat Step1 and Step2, then we can get a series of values for X . This way we say that we generate a random variable X . Let FX (x ) be the CDF of X . From the description, we have X = F −1 (U ) and U = F (X ). We have FX (x ) = P (X ≤ x ) = P F −1 (U ) ≤ x = P (U ≤ F (x )) = F (x ) Thus the generated random variable X follows the desired CDF F (x ). M. Chen ([email protected]) ENGG2430C lecture 9 18 / 20 Generating Random Variable: Example* Describe how your procedure can be applied to simulate an exponential r.v. with parameter λ = 10. Justify your answers. Deﬁne the exponential r.v. as Y , so FY (y ) = 1 − e −10y . To generate Y , the procedure is Step 1: Generate a value of U using the computer, say the value is u . Step 2: Compute the value y which satisﬁes that u = 1 − e −10y , called 1 ychosen . ychosen = F −1 (u ) = − 10 log (1 − u ). Step 3: Repeat Step1 and Step2, then we can get a series of values ychosen . These values are random numbers drawn from the exponential distribution with parameter 10. M. Chen ([email protected]) ENGG2430C lecture 9 19 / 20 Thank You Reading: Ch. 3 of the textbook. Next lecture: Inequalities and Law of Large Number. M. Chen ([email protected]) ENGG2430C lecture 9 20 / 20...
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