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Exercise 1 A popular text on simulation theory states that if U~ U(0, 1), then log (U) (6) has an exponential distribution with parameter X.

Exercise 1 A popular text on simulation theory states that if U~ U(0, 1), then log (U) (6) has an exponential distribution with parameter X.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 1
1 F-¹(x) = -log(1-x) So, by the theorem just stated, if U is uniform on (0,1) then -log(1-U) has the exponential distribution. (4) 5 Exercise 1 A popular text on simulation theory states that if U~ U (0, 1), then -log(U) (6) has an exponential distribution with parameter A. Note that the 1-U in Equation (5) has been replaced by U in Equation (6). Briefly explain why this is a valid substitution.
Transcribed Image Text:1 F-¹(x) = -log(1-x) So, by the theorem just stated, if U is uniform on (0,1) then -log(1-U) has the exponential distribution. (4) 5 Exercise 1 A popular text on simulation theory states that if U~ U (0, 1), then -log(U) (6) has an exponential distribution with parameter A. Note that the 1-U in Equation (5) has been replaced by U in Equation (6). Briefly explain why this is a valid substitution.
Recall that the cumulative distribution function (cdf) associated with a random variable X is the function F(x) = P(X ≤ x). The following theorem is based on the calculus of transformations: Theorem 1.1 If a random variable X has cdf F, and if U is a random vari- able uniform on (0,1), then the random variable W = F-¹(U) has the same distribution as X. Example Suppose one wishes to generate random values from an exponential distribu- tion. That is values of a random variable with pdf f(x) = Ae¯ª, for x ≥ 0 where is a positive constant. One computes F(x) = ª* \e¯`dt = 1 − e¯λ² Some elementary algebra gives (1) 1
Transcribed Image Text:Recall that the cumulative distribution function (cdf) associated with a random variable X is the function F(x) = P(X ≤ x). The following theorem is based on the calculus of transformations: Theorem 1.1 If a random variable X has cdf F, and if U is a random vari- able uniform on (0,1), then the random variable W = F-¹(U) has the same distribution as X. Example Suppose one wishes to generate random values from an exponential distribu- tion. That is values of a random variable with pdf f(x) = Ae¯ª, for x ≥ 0 where is a positive constant. One computes F(x) = ª* \e¯`dt = 1 − e¯λ² Some elementary algebra gives (1) 1
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